Consider the following fields:
1) $\mathbb{C}$ the complex numbers
2) $\overline{\mathbb{Q}}_p$
3) $\mathbb{C}_p : = \hat{\overline{\mathbb{Q}}_p}$
They are all the same cardinality, algebraically closed, and of characteristic 0. Therefore they are all isomorphic as fields. However: $\overline{\mathbb{Q}}_p \to \mathbb{C}_p$, since it is the topological completion, and is not surjective. Is there something wrong with this argument or not?
Edit (To make my question more clear): Is $\overline{\mathbb{Q}}_p \to \mathbb{C}_p$ surjective? Or can there exist embeddings of fields into isomorphic fields which are not surjective?
Nothing is wrong with the argument. These objects are isomorphic in the category of fields and not in the category of topologial fields (or even the category of topological spaces).