'Monarchia' is the Anglicized Greek word for 'First Principle'. In many ways, this was a fundamental question in Greek philosophy. The pre-Socratics tried to figure out the fundamental building block of the world (and perhaps, through such, discover the nature of deity). Thales said it was water, Anaximander said it was the Monad (an indescribable, theoretical, particle that precedes all), and Anaximenes said it was aether, a kind of purified air. Democritus and the Atomists believed in the 'Atom', the word of the undividable, in a materialist kind of metaphysics. Heraclitus believed in Change, the endless flux of things, materialized as Fire, and hence the soul of the world was a divine fire. Empedocles believed in the ultimate flux of love and hate, attractive and repulsive principles that made up life. Parmenides believed that all diversity was a lie and that reality was only One, all difference was an illusion.
The above is to give you a taste of the diversity of thought in Hellenic philosophy. Lest philosophy be completely abstract, look at how many of these philosophers found correlates in material existence or Human life. While the creativity and lively diversity of the above is distinctly Greek, the question itself is not fundamentally one trapped in Western civilization. This is assuming the Greeks are really even "Western", which is more of a teleological convention to explain the global dominance of Rome and, then later, Western Europe.
But barring that discussion, many peoples have asked similar questions, but in distinctly different terms. While the above Greek philosophers looked for a kind of de-personalized divine essence, other nations put the question in what people might consider more traditional "religious" language. While most western Africans were polytheistic, many believed in the notion of a Great god that was above all, but far and away. Different American Indians believed in some kind of spirit that was primeval force of creation. The Chinese believed in the supremacy of "Heaven" over all the forces of creation. Even more conventional Greeks believed in the supremacy of Fate, even above Zeus (if not synonymous with him) and the other gods.
All of this is a question of 'monarchia' and there's something Human about the search for such. Of course, the question is general and broad and not all formulations are helpful or are necessarily related to the truth. Aristotle's notion of the Prime Mover considers all of life within the sphere of momentum and can be a tool in the mechanization of all things. The great god of western Africans maintains, within it, a sense of space, as the great god needs mediators (and hence why many gods are posited and worshiped) as he is so far away. Greek philosophy's search for a prime material makes divinity as the life-blood of creation, and circumvents any question of separation (what we might say is the 'holiness' of divinity).
As per the Scriptures, the Creation testifies to this truth ("For since the creation of the world [God's] invisible attributes are clearly seen...even His eternal power and Godhead"), but we don't know the answer through our own poor estate in the bondage of death and corruption. Christ reveals the fullness of God, as true God of true God, the Father that none has ever seen or known. The Scripture testifies and reveals the fullness of the Godhead. We thus go and make disciples of all nations, baptizing in the name of the Father, the Son, and the Holy Spirit.
But of course, Trinitarian theology, while a necessity, has become a locus of the question of monarchia. That is, what do we mean by 'Father', 'Son', and 'Holy Spirit'? Is there an ordering of divinity? Are there three Gods? Is there one that is more God than the other? How exactly do they relate to one another? How do they relate to us? These sorts of questions, and more, plague the Christological debates of the Church throughout all eras, but particularly through the first seven centuries of the Christian Church.
And lest this be considered a bizarre, and insane, exercise of speculative theology, like the Greeks above, one must consider the nature of the Scripture. If the Scripture is truly the Word of God in textual formula, it is a true revelation from God's mouth about Himself and the Creation He has made. It is within such boundaries that many argued, though perhaps, at times, stepped out of boundaries and were, if not nearly, unmoored from divine revelation. Jesus Christ is the anchor of all these discussion and hence discussions of Christology almost always involved the Godhead.
I don't intend to give a history of this theology, but instead I thought I'd lay out my position and why it relates to the above question of monarchia. But before I proceed, I will make an assertion that I'm taking as the bedrock of my statements: How we address the question of 'monarchia' is the fundamental preposition from which we do theology outwardly. That is to say, theology is about seeking God, not as a science, except in the sense that it involves knowledge. Thus, theology begins and ends in who God is. Thus the question of Trinity, monarchia, and even Christology, deeply impact how we think about how we relate to God. Of course, knowledge does not preclude a relationship. Rather, it's the other way around usually. We are in relationship, and we seek to understand 'who' our Friend is.
Anyway, the question of monarchia must reside in the fact of the 'monos' part. As per the Shema, we must recognize that our God is One. Of course, a monarchia does not equate to a monad, as per Judaism and Islam's polemics against Christians. In other words, acknowledge a first principle, the Godhead, does not mean Trinitarian automatically equals tri-theism, belief in three gods. Judaism and Muslims critique that Christians are polytheists and pagans is not true. But the criticism ought to warn how we think about this.
Modern day Social Trinitarianism comes close to embodying the tri-theism that the above critics decry. By referring to God as a Communion, which Scripture never does, it seems to make God divisible. It says, implicitly if not explicitly, that each Person of the Trinity has its own individualizable existence.
I am not disagreeing with the language of Personhood, i.e. the Father is Person, Son is Person, and Holy Spirit is Person, Three Persons as One. What I do take issue with is that Person might be used to similarly to Human Person, which connotes a kind of individual conscious center. One sees this in constructions of how God decided to save Humanity, with the Father making an agreement with the Son to save Mankind through Incarnation (pactus salutis). This construction, if literal, makes it seem that the Son and the Father had to come to agreement, implying the Son could've said 'No'(!) and Humanity would've been doomed.
Here's my proscription, take it or leave it: The New Testament, particularly St. Paul, ascribes fully divinity to the Son and the Spirit, but almost always refers to the Father as 'God'. We always hear "God the Father and Jesus Christ" (Christ, meaning 'anointing', always implies the Spirit with the Son). Without denying the titular God to the Son or the Spirit, it's worth considering the fact that the Father, whom the Son and Spirit reveal, nominally bears the title, in the non-descriptive, God.
While the Arians were wrong, they keep us honest: what are we to do with this language? We have to recognize and ascribe monarchia to the Father, as the Fount of Divinity. However, contrary to the Arians, of whatever era or flavor, this does not diminish the deity of the Son or the Spirit. As per the Trinitarian formula, grounded in Scripture, the Father begets the Son in the Spirit, and the Father processes the Spirit through the Son. As God is timeless, these acts are not chronological, but eternal. There was never a time when the Father was not Father, and to be Father requires the begetting and processing of the Son and Spirit. They are not ordered in a hierarchical form. The Father speaks his Word through Spirating, and Spirates through the speaking of His Word.
However, this seems to imply a whole lot of passivity in the Son and the Spirit, perhaps revealing a kind of lesser status. One solution to this is to collapse some of the distinctions between the Three so that we don't speak so clearly between Father, Son, and Spirit. Some approaches to this are more problematic than others. Sabellianism makes each Person only a mask that true God wears for a time. Even some of theology in the West has seemed to posit some Fourth behind the existence of the Three, as there is a Divinity, a true and pure Godhead, behind the Father, Son, and Holy Spirit. If we are to say, with Tom Torrance, that there is no God behind Jesus Christ, than we can never go this way. If Jesus (the Son) is Christ (in the Spirit), the Son of God (begotten of the Father), than we must confess a Trinity. And if we confess the eternal everlasting Kingdom of Jesus Christ, then these distinctions of the persons are real.
But Torrance, and some like him, made the mistake of saying that Father, Son, and Spirit is God, which almost invites a confusion of the Persons or a collapse into a Fourth. There may be no God behind the back of Jesus Christ, but there is One above Him, namely His Father. And this makes sense of the Biblical language, even of Christ Himself, without collapsing into abstraction.
One might hear the word 'perichoresis', the 'inner dance' of the Triune life (i.e. Father and Spirit in and with Son, the Son and Spirit from the Father etc etc.), which is correct to a limited extent. But it's very easy to run away with this into unbiblical abstraction. Yes, the Divine life is indivisible and always One, but this does not erase the clear distinction (not divisions or otherness) in the Bible, both Testaments. The Father never prays to the Son, the Son is the one who does incarnate, and the Spirit never speaks. In other words, the above kind of Theology borders on a kind of functional Sabellianism, which drew the question "Then why Three? Why not Four? Ten? Seventy?". So we can ask, "Why does the Son pray to the Father?" "Why doesn't the Spirit incarnate?" etc etc. We see thus in the, well-meaning but theologically confusing, book "The Shack".
We must, if we wish to be loyal to the words of Scripture, ascribe monarchia to the Father. However, as I mentioned before, this does not mean a kind of passivity. While the Son is begotten is put in passive tense, we must consider that the Son is always is doing what the Father is doing (Jn 5:17). That is to say, this passive phrase is an active tense. Same goes for the Spirit, in His own way. While we might say, on one hand, that the Son and Spirit's Godhood is derived (in Their Begottenness and Spiratedness, respectively), we can say that the Son is God in Himself. Why? Because He is always the Son. In another phrase, even if the Father does not Speak His Word to Creation, and Breathe out upon it, they still remain within the Father.
The Father is always with the Son and the Spirit, and in this way we might speak of perichoresis, as they cannot be considered apart from one another. As Gregory Nazianzus, that profound and blessed theologian, a friend of God, said, in so many words, 'I cannot think of the One without thinking of the Three, and I cannot think of the Three without thinking of the One'. This does not mean confusion or blurring, but that the Three Persons share a Life, eternally manifested through the Father's begetting and processing. The monarchia of the Father is true and right, in the same way that we understand that the Son is the One who Incarnates, and the Spirit inspires.
While this may seem important in an abstract sense, it is also acutely important for a practical sense. In a strictly analogous way, God's actions internally are connected to how He relates to His creation They are not separable, as per some apophatic theologians who almost border agnosticism of the True God behind how God appears to us. Even Luther trembled before, and tried to put out of his mind, 'Deus Absconditus', the 'Hidden God', that Torrance rightly criticized.
If our God is the God who's life is distinctly reveal, we might be able to rejoice in the fact that we know the Creator, something awe-inspiring and shocking. The above is how we ought to understand what St. John meant when he said "God is love". But this kind of love allows for distinction without division or difference. In the Son, God became Man so Man might become gods (as per Athanasius the Great). Thus the life of God is opened up to Humanity in a way that we might truly bear His image, in a way that God the Son did, by nature, when He took on sinful Human flesh. We are promised to be partakers of the Divine Nature, as per St. Peter. In short, we actually become close to God, live in His life, and yet remain ourselves, without losing ourselves as per Pantheistic theologies.
Some of the above might seem technical, and there might be additional problems to discuss or unwind. The above is not comprehensive, but an introduction to why thinking about the Trinity is not unimportant or uncomplicated or just irrational. We are dealing with the fact that God has spoken, speaks, and will continue to speak to us, His creation, which He has redeemed through the blood of His Son, yea, redeemed by God's own blood (Acts 20). If we don't consider the question of 'monarchia' then we're prone to assume poorly and end up as functional Unitarians or Sabellians. We will miss the fact that we actually can become children of God, and yet remain the individual Human persons that God made us to become.