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> If we do it your way and Complex Numbers become just fucking dust, then *all* of our math becomes garbage and we're totally fucked. The thing is that it is *entirely possible* that we're just completely fucking wrong about our axioms - they're *axioms* after all. We'd like very much for as much math to stay intact if we have to give up on an axiom, and we build it the way we do because we're more willing to accept "Real Numbers are garbage" than "no there aren't actually Natural Numbers."
of course, natural numbers are just complex numbers of a certain kind: complex numbers with angle = 0 / imaginary component = 0 / 2nd column = 0 /etc and which the "real" part / magnitude >= 0
ie if you grant complex numbers you can easily define reals / natural numbers. ie complex numbers are more fundamental.