[articulatepang]

As you say, "the fundamental theorem of algebra relies on complex numbers" gets to the heart of the view that complex numbers are the algebraic closure of R.

But also, the most slick, sexy proof I know for the fundamental theorem of algebra is via complex analysis, where it's an easy consequence of Liouville's Theorem, which states that any function which is complex-differentiable and bounded on all of C must in fact be constant.

Like many other theorems in complex analysis, this is extremely surprising and has no analogue in real analysis!