Do the properties of metric completeness and algebraic closure characterize the complex numbers up to isomorphism?

Question

If not, are there any such properties?

Answer 1

I cannot find the topic where I thought I saw this a couple days ago, so here it is:

Short answer: No. There are infinitely many fields which are complete and algebraically closed and not isomorphic to $\Bbb C$.

You want a minimal algebraically closed, complete, archimedean field containing $\overline{\Bbb Q}$ or $\Bbb R$, or something of the like. You could also use locally compact, archimedean, algebraically closed and drop the minimal assumption.