My apologies for misunderstanding the question.
But I did write at some length above about the multipersonal distinction of a self-begetting and self-begotten singular ultimate entity. (Mostly setting aside the topic of the 3rd Person as requiring further discussion.)
The difference is perhaps in the direction of the logical approach. When I go through the metaphysical analysis, I discover first that I should believe in one ultimate Independent Fact (whether naturalistic or supernaturalist, theistic or atheistic) upon which all reality must depend for existence.
Then I discover that I should believe this IF to be rationally active (or actively rational, to put it the other way around). I thus accept theism to be true and reject atheism. But naturalism or supernaturalism might still be true.
The next issue I come to doesn’t solve for naturalism or supernaturalism either way (that comes later), but involves how a rationally active ultimate existence ultimately exists: by rationally acting. (The alternative, that this Independent Fact exists merely statically, not even self-causingly, runs into conflict with prior portions of my analysis, for reasons I cannot easily summarize here.) Put a little overbriefly, though, this self-generating action of God results in a paradoxical distinction that would be unique to the finally active cause of all existence: there is a conceptual action of cause and effect which, at this ultimate and finally irreducible level of existence, are effectively the same thing. God is both self-begetting and self-begotten. But this doesn’t mean that God, being self-begotten, is not God!–anymore than God, being self-begetting, would somehow thus not be God.
Consequently, since God, as an active sentience, is a personal entity (though not in the limited way that I am a personal entity, but upon which, or upon Whom, rather my own rational characteristics ultimately depend for existence), I thus am led to simultaneously affirm that God Self-begetting is a Person, and God Self-begotten is a Person.
(Note: at this point I’m working from some unpublished material that would otherwise fit into Section Three of SttH, the free postings of which are linked to in all my signatures.)
So far this might only be a modalistic doctrine. What God most fundamentally acts to do, in self-generational existence, must be fully and completely Himself; but this might only mean that we should treat this characteristic of God (a Person is self-begetting, a Person is self-begotten) as something of a useful legal fiction, the way we might consider a self-existent equation to be two distinct formulas because the formulas (though they are ultimately the same) ‘look’ different. Thus for certain purposes we might use or refer to the formula on the left side of the equal sign, while for other purposes we might be better served by referring to the formula on the right. The statement of principle would in either case be ultimately the same, but we might find different valid uses for different expressions of the statement. To this extent, enriching my perception of God by recognizing a ‘unity in multiplicity’ might be quite useful to me (the subjective observer); but by itself this doesn’t make it necessarily more than a convenient description.
However, now we come to the philosophical problem of consciousness. We have discovered that it is inconsistent to claim that a person is ‘conscious’ without some existent distinction for the person to be perceptive of for purposes of self-identification.
The problem then becomes this: if I take my own rationality seriously (according to a previous analysis which I have not summarized here), then I will fetch up sooner or later at the necessary existence of a sentient Independent Fact: God. It would be inconsistent (I agree with the atheists here) to say that God has these properties and yet is not conscious at that state of His existence where nothing else has been created. And I further agree (again with the atheists) that without a distinctive difference of states, it is nonsensical to say that God could be ‘conscious’. If I deduct that God exists, however, (with an inference such that I cannot feasibly believe God doesn’t exist without undermining any argument I might make on any topic), then I should be expecting some kind of real Personal distinction to obtain in the self-existence of God.
And the begettor/begotten distinction satisfies this requirement in the most basic manner possible; for differentiation requires some type of action by the IF, and there can be no more basic action than self-generation.
Thus I conclude that the most fundamental action of God, God’s self-generation, eternally introduces into His own most basic level of reality a true distinction of some sort; one which is intimately connected to the relationship between God’s action of self-generation and the result of self-generation which is He Himself God.
The simplest possible way of stating this would be: God the Begettor is in some true sense one distinctive Person, and God the Begotten is in some true sense another distinctive Person. The Father/Son imagery turns out to be increasingly more accurate. God, at His most fundamental level of reality, is first and foremost a Unity of Persons–one distinctively Begetting and one distinctively Begotten, both of them constituting a common ‘substance’ (so to speak) of existence.
Can I mentally picture this? No, and I doubt there is any properly full analogy for it either. But (just as in quantum physics, for example) I am not worried about a lack of totally accurate mental imagery, as long as the underlying precepts (the “logical math”, so to speak) remain self-consistent. The Father/Son imagery, as far as I can tell, is adequate: God the Father eternally begets God the Son, Who eternally submits in self-consistency back to the Father. The Son is of one mind with the Father and does the Father’s will, and indeed does nothing except what the Father does, being the very action of God Himself. The Son may be said to be dependent upon the Father, as God is dependent upon Himself for His very existence. The active sentience of God, however, requires for God’s own self-consciousness, that this unity of Persons be true not only modally (it would be at least that), but as a real distinction of Persons, at the level of God’s eternal self-sufficient existence.
(I’ve had to skip over a few important topics, such as why distinction-of-an-other-for-self-consciousness would also be true of God and not only a limitation of derivative sentience such as my own. Consider such things to be topics for further discussion. 
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I’ll try to post up a new version of it later, but first I’m going to look around and try to figure out if I somehow sent it elsewhere on the forum by accident… (is that even possible??)
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) It’s “very God of very God”, as one creedal translation puts it.