I know that$\mathbb R$ is - up to a unique isomorphism of ordered fields - the unique complete ordered field. Is there a similar characterization of the complex numbers? I guess that $\mathbb C$ can not simply be defined as a field with particular properties, but that additional structure must be part of the definition.$^1$
Anyways, I am sure that many people have thought about this issue and that there are several possible approaches. I would like to learn about them.
$^1$ E.g. two functions from $\mathbb C$ onto $\mathbb R$ (the projection onto the real part and the projection onto the imaginary part).