> In particular, they arose historically as a tool for solving polynomial equations.

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.