@jeffcliff @plotinusgroyper Oh, this is a strong misunderstanding of how these things are built but that's normal for someone who hasn't studied it extensively.

The standard way we construct the Complex Numbers *normally* goes kinda like this:

1) Start with the Natural Numbers (built somehow using set theory).
2) Build the Integers by giving the Natural Numbers a group structure (additive inverses, a zero, and addition)
3) Build the Rational Numbers by giving the Integers a field structure (multiplication, multiplicative inverses, and a one.)
4) Build the Real Numbers by Dedekind Cuts or similar
5) Build the Complex Numbers as the Algebraic Closure of the Reals (Add in all of the roots of polynomials which have real coefficients).

We don't "start" with the Complex Numbers, we actually do build them from other parts. I don't actually see how you'd build the Complex Numbers *other* than as the Algebraic Closure of the Reals.