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!
That is how they started, but mathematics becomes remarkable "better" and more consistent with complex numbers.
As you say, The Fundamental Theorem of Algebra relies on complex numbers.
Cauchy's Integral Theorem (and Residue Theorem) is a beautiful complex-only result.
As is the Maximum Modulus Principle.
The Open Mapping Theorem is true for complex functions, not real functions.
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Are complex numbers really worse than real numbers? Transcendentals? Hippasus was downed for the irrationals.
I'm not sure any numbers outside the naturals exist. And maybe not even those.