Why don't we have an intrinsic representation of complex numbers? In many ways that I probably don't have to list here the complex numbers $\mathbb C$ arise very naturally, many times they seem to be the more natural set of numbers to work with than real numbers $\mathbb R$.
However whenever we do work with them, it seems that we always use the crutch of representing them as a pair of real numbers, here just a list of a few examples:
$$a+ib \qquad re^{i\varphi} \qquad \begin{bmatrix} u & -v \\ v & u \end{bmatrix} \qquad r+sx + (1+x^2)\mathbb R[x]$$ 
However natural they are, it seems we don't really have an "intrinsic" representation of the complex numbers that we can work with - at least not that I am aware of. So this leads me to the following questions:
Why is that the case? Or is there anything closer to an intrinsic representation than anything similar to the above examples, i.e. that does not use just a pair of real numbers?
 A: So let's start with the fact that in mathematic everything is built on top of something else. Complex numbers are built on top of reals. Reals on top of rationals, rationals on top of integers, integers on top of naturals, naturals on top of sets. Wherever you start, as long as you go deep enough you will eventually reach sets, which are not really defined, they just are. They are supposed to obey some axioms, yes, but they are not constructed from anything.
So I guess you can say that sets have an intrinsic representation. But nothing else in maths. Well at least in ZF set theory (we do have to assume something, right?).
Now you've said that complex numbers are defined in terms of reals and that is not an intrinsic representation. But do reals have an intrinsic representation? Reals are constructed from rationals as equivalence classes of Cauchy sequences (there are other equivalent constructions, it doesn't matter). So each of these symbols $1, 7, -4, \frac{1}{3}, \sqrt{2}, \pi$ is just a convention. It is a label, it is not a real number. The real real number is an equivalence class of some Cauchy sequence. And those labels are far from perfect, there are not enough words (in a natural language, which I assume is at most countable) to construct every possible real number. We are just so used to these few common labels being used every day that they seem so natural to us. But let me remind you that few hundreds years ago that wasn't the case. For example Diophantus considered negative numbers as "useless, meaningless, and even absurd".
Anyway, in a similar manner you can invent some arbitrary convention, some "labels", that will represent complex numbers in a new way. But this feels like cheating, because the essence stays the same: complex numbers are pairs of reals, you can't avoid that. Just like you can't avoid the fact that rationals are inside reals, you can only hide that.
That being said, you can argue that there is the axiomatic approach to reals. But as I've mentioned earlier, axioms do not define anything. At most they define a behaviour that something is supposed to have. You still have to construct at least one object that satisfies your axioms.
But I suppose you could ask: is there an axiomatic approach to complex numbers? Sure, for example as an algebraic closure of reals. But without using reals? I've never seen it. Note that the existence of reals may be hidden somewhere. I've seen once a paper that dealt with a classification of path connected topological fields. But you can't really talk about "path connected" without reals, so yeah...

Why is that the case?

This is a rather philosophical question. Mathematics (and any other science) doesn't really answer "why" questions but rather "how" questions.
