I read in the post Why worry about the Axiom of Choice ? that the existence of isomorphisms between $\overline{\mathbb{Q}_p}$, $p$ any prime, and $\mathbb{C}$, makes some worry about the Axiom of Choice. One can find another interesting discussion in Are $\mathbb{C}$ and $\overline{\mathbb{Q}_p}$ isomorphic ? (cf. also the references to Deligne's work on the Weil conjectures). My question is: are there other (highly) unexpected isomorphisms between at first sight unrelated fields ?

Thanks !