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 $C$ and $\bar{Q_p}$. The proof of said isomorphism runs as follows. Both $C$ and $\bar{Q_p}$ have transcendence bases, $S$ and $T$. So $C\simeq \bar{Q(S)}$ and $\bar{Q_p}\simeq \bar{Q(T)}$. But $C$ and $\bar{Q_p}$ have the same cardinality, and hence, so do $S$ and $T$. Therefore, $Q(S)\simeq Q(T)$ and, from there, $C\simeq \bar{Q_p}$.

For myself, this proof is quite convincing. Recently, Torsten Ekedahl expressed his opinion to the contrary, that 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?