It seems to me (intuitively) that there should be no other fields whose algebraic closure is $\mathbf{C}$, even though I have no reason to believe it. The facts I've been using to formulate an argument are $[\mathbf{C}\mathbin{:}\mathbf{R}]=2$ and $\mathbf{R}$ is the only field with the usual analytic properties. I mean, it seems that for the complexes to be even defined we need to reference the analytic properties of the reals. Also, we know that such a field would have to be uncountable, right?
This question might end up being trivial, but any information or references would be appreciated.