There is a famous passage on the third page of Deligne's second paper on the Weil conjectures where he expresses his dislike of the axiom of choice, as manifested in the isomorphism between $\mathbb{C}$ and $\overline{\mathbb{Q}}_p$. The proof of said isomorphism runs as follows. Both $\mathbb{C}$ and $\overline{\mathbb{Q}}_p$ have transcendence bases, $S$ and $T$. Then $\mathbb{C}\simeq \overline{\mathbb{Q}(S)}$ and $\overline{\mathbb{Q}}_p\simeq \overline{\mathbb{Q}(T)}$. But $\mathbb{C}$ and $\overline{\mathbb{Q}}_p$ have the same cardinality, and hence, so do $S$ and $T$. Therefore, $\mathbb{Q}(S)\simeq \mathbb{Q}(T)$ and, from there, $\mathbb{C}\simeq \overline{\mathbb{Q}}_p$.

For myself, this proof is quite convincing. Recently, Torsten Ekedahl expressed his opinion to the contrary, and this led to the following exchange:

Why worry about the axiom of choice?

So I wondered about other expert opinions on this matter. Do you find the isomorphism unbelievable and, if so, why?