I'm afraid this was a bit too long for a comment.

It seems that people don't object too much when we assert that $\mathbb{C}$ and $\overline{\mathbb{Q}_p}$ are isomorphic as sets or as abelian groups. Somehow the use of choice when establishing a ring-theoretic isomorphism bothers mathematicians much more, and I suspect it is because the implications are more at odds with intuition we build up from considering finite extensions of fields, or geometric structure that is normally attached to the fields. When considering purely ring-theoretic maps, we are still forgetting a lot of structural baggage, e.g., an affirmative answer to the similar question about existence of an embedding of fields $\mathbb{C}(t) \hookrightarrow \mathbb{C}$ throws away anything we know about genus zero curves.

I would personally answer your question with "yes" although I would not argue the point with much conviction. I would be interested to know if there were a logical way of chopping up choice so that the isomorphisms of sets and groups were okay but the isomorphisms of rings were not.