It's not like I have a real answer, of course, but something flipped inside of me after hearing the following story by Aaronson. He is asking[0], why quantum amplitudes would have to be complex. I.e., can we imagine a universe, where it's not the case?
> Why did God go with the complex numbers and not the real numbers?
> Years ago, at Berkeley, I was hanging out with some math grad students -- I fell in with the wrong crowd -- and I asked them that exact question. The mathematicians just snickered. "Give us a break -- the complex numbers are algebraically closed!" To them it wasn't a mystery at all.
Apparently, you weren't one of these math grad students, and, to be fair, Aaronson is starting with the question that is somewhat opposite to yours, but still, doesn't it intuitively make sense somehow? We are modeling something. In the process of modeling something we discover functions, and algebra, and find out that we'd like to use square roots all over the place. And just that alone leads us naturally to complex numbers! We didn't start with them, we only imagined an algebra that allows us to describe some process we'd like to describe, and suddenly there's no way around complex numbers! To me, thinking this way makes it almost obvious that ℂ-numbers are "real" somehow, they are indeed the fundamental building block of some complex-enough model, while ℝ are not.
Now, I must admit, that of course it doesn't reveal to me what the fuck they actually are, how to "imagine" them in the real world. I suppose, it's the same with you. But at least it makes me quite sure that indeed this is "the shadow of something natural that we just couldn't see", and I just don't know what. I believe it to be the consequence of us currently representing all numbers somehow "wrong". Similarly to how ancient Babylonian fraction representations were preventing ancient Babylonians from asking the right questions about them.
P.S. I think I must admit, that I do NOT believe real numbers to be natural in any sense whatsoever. But this is completely besides the point.