"Why" is $[\mathbb{C}:\mathbb{R}] < \infty$?

Obviously this question is a little open-ended.

A lot of complex analysis seems to work primarily because we can view $\mathbb{C}$ as a finite-dimensional $\mathbb{R}$-algebra, and apply analytic and geometric ideas which work only (or at least best) in finite-dimensional real space.

When we consider most other fields that we come across in practice (for instance, $\mathbb{F}_q$,$\mathbb{Q}_p$, or $\mathbb{Q}$) generally their algebraic closures are infinite-dimensional extensions.

Is there any intuitive reason why the way we construct $\mathbb{R}$ would suggest that we were producing a field whose algebraic closure was a finite-dimensional extension? Does such a construction generalize to other fields in any way?

I can't give you an "intuitive" reason why $\Bbb C$ is a finite dimensional $\Bbb R$-algebra, but I can point you toward a generalization of this fact : $\Bbb R$ is a real closed field, and if $F$ is any real closed field, then $F[\sqrt{-1}]$ is algebraically closed.

EDIT As @omar, @Grigory and @Hurkyl pointed out in the comment section, the relevant material is the Artin-Schreier theorem.