I've decided to attempt the entire Rudin sequence in a single 6 month period, because I'm insane. Rudin spends very little time on foundational matters, and that bothers me, it makes the subject of analysis feel less philosophically sound. So I've been adding the missing axioms and proofs as I go.
I've defined the real numbers as the unique model (up to isomorphism) of the second-order theory over the language $\langle+,\cdot,<\rangle$ or $\langle+,<,\cdot,0,1\rangle$ given by the axioms:
$\forall x\forall y\forall z[(x+y)+z=x+(y+z)]$
$\forall x\forall y \forall z [(x\cdot y)\cdot z=x\cdot (y\cdot z)]$
$\forall x\forall y(x+y=y+x)$
$\forall x\forall y(x\cdot y=y\cdot x)$
$\forall x\forall y\forall z(x\cdot(y+z)=(x\cdot y)+(x\cdot z))$
$\exists x\forall y(x+y=y)$ or $\forall x(0+x=x)$
$\forall x\exists y\forall z[x+y=z\implies\forall w(w+z=w)]$ or $\forall x \exists y(x+y=0)$
$\exists x\forall y(x\cdot y=y)$ or $\forall x(1\cdot x=x)$
$\forall x\forall y\forall z[(x+y=x\land x\cdot z=x)\implies y\ne z]$ or $0\ne 1$ (not sure if this one is necessary)
$\forall x[\exists y(x+y\ne y)\implies\exists y\forall z((x\cdot y)\cdot z=z)]$ or $\forall x[x\ne0\implies\exists y (x\cdot y=1)]$
$\forall x\forall y\forall z(x<y\implies x+z<y+z)$
$\forall x\forall y\forall z((x<y\land y<z)\implies x<z)$
$\forall x\forall y(x<y\lor y<x\lor x=y)$
$\forall x\forall y\forall z[(x<y\land\exists w(w+z\ne w))\implies x\cdot z<y\cdot z]$ or $\forall x\forall y\forall z((x<y\land z>0)\implies x\cdot z<y\cdot z)$
$\forall P[(\exists xPx\land\exists y\forall x(Px\implies x<y))\implies\exists y\forall x((Px\implies x\le y)\land\forall z(z<y\implies\exists x_1(Px_1\land z<x_1)))]$
I picked these axioms over say, the Tarski axioms, because the standard first-order characterization can be easily recovered by replacing the second-order completeness axiom with its first-order equivalent. That way, I can keep pretending that I'm not using SOL as long as I'm subtle with my wording (technically, this is called "lying," but the belief that analysis is somehow a first-order endeavor, and that first-order logic is somehow the "true" logic, is so deeply ingrained in the subject that I've given up correcting people.)
Now I want to define the complex numbers. I can think of several ways to do this, but all of them make explicit reference to the real numbers, and that bugs me. You shouldn't need to define the very specific Dedekind complete ordered field of real numbers to define the complex numbers; it should be possible to recover it from an adequate single-sorted characterization of the complex numbers as a field (up to isomorphism.) Yet every set of axioms I can find or invent ends up directly referencing either $\Bbb R^2$, by way of needing a predicate for "real" or implicitly defined projection functions (e.g.$\forall x\exists!y\exists!z\cdots$ := "$x=y+iz$"), or the topology on $\Bbb C$ (the second-order characterization of the Euclidean topology on $\Bbb C$ suffices to characterize all analysis-relevant properties.)
We don't usually define the real numbers to by a two-sorted theory describing such-and-such ordered field containing $\Bbb Q$. Instead, all of the expected properties just arise from the ordering on the field as though by miraculous coincidence. Why should the complex numbers be different?
How can I "properly" axiomatize the complex numbers (over $\langle+,\cdot\rangle$, $\langle+,\cdot,0,1\rangle$, or $\langle+,\cdot,0,1,i\rangle$) so that the associated properties (the ordering of the reals, conjugation, the standard topology, etc.) are implied by the axioms without being stated outright? And how can I do it in a way that let's me pretend that SOL exist in different universe from analysis?
In the presence of the axiom of choice, $\mathbb{C}$ is the unique algebraically closed field of characteristic zero and size $2^{\aleph_0}$. There is no first-order characterization of "size $2^{\aleph_0}$" (by downward Lowenheim-Skolem), and this isn't a very useful characterization anyway, because it doesn't let us do any topology or analysis. $\overline{\mathbb{Q}_p}$, for example, is also such a field and carries a quite different topology even if, in the presence of choice, it must be abstractly isomorphic to $\mathbb{C}$.
Using the classification of locally compact fields, $\mathbb{C}$ with the Euclidean topology can also be characterized as the unique locally compact topological field which is 1. algebraically closed and 2. connected. This is more satisfying but also requires some pretty nontrivial tools and ultimately it's not clear to me how valuable it is. In practice we really do care about complex conjugation and about the way the real numbers sit inside the complex numbers and I don't see a compelling reason to try to avoid this. This is also a characterization and not a construction, one still has to construct such a topological field through other means.
If you are specifically interested in understanding how the complex numbers arise in quantum mechanics, you may be interested in Vicary's Completeness of dagger-categories and the complex numbers which gives an axiomatization of the entire dagger category of finite-dimensional complex Hilbert spaces (which gives us $\mathbb{C}$ equipped with complex conjugation, and hence gives us both $\mathbb{C}$ and $\mathbb{R}$ simultaneously).