First, let me observe that it is consistent with ZF$\mathsf{ZF}$ + DC$\mathsf{DC}$ that there is no such isomorphism. (This follows from this answer of mine.) However, as I commented on Torsten's post, the existence of such an isomorphism is a relatively harmless since one can force the existence of such an isomorphism without adding new points to C$\mathbf{C}$ or Qp$\mathbf{Q}_p$. Consequently, any purely field-theoretic fact that can be proved using this generic isomorphism can also be proved without (usually with more work). Since forcing is not widely understood, I will explain this in terms of sheaves instead. (If you're more familiar with forcing and you don't care about sheaves, simply observe that the poset P$P$ below is countably closed and ignore the rest of this post.)
Let P$P$ be the poset of field isomorphisms p:A→B$p:A\rightarrow B$ where A$A$ is a countable subfield of C$\mathbf{C}$ and B$B$ is a countable subfield of the algebraic closure of Qp$\mathbf{Q}_p$, and p ≤ q$p \le q$ iff p ⊇ q$p \supseteq q$ (i.e. q$q$ is a restriction of p$p$). This ordering is slightly counterintuitive, but it is more convenient than the opposite. The poset P$P$ can be viewed as a category where there is one and only one arrow between any two objects p$p$ and q$q$ iff p ≤ q$p \le q$. The poset P$P$ then becomes a Cartesian category where the terminal elementobject is the isomorphism between the two copies of Q$\mathbf{Q}$ in each field, and the product of p$p$ and q$q$ are is the intersection of the (graphs of) p$p$ and q$q$.
There are many Grothendieck topologies that one could define on P$P$. The relevant one for our context is the smallest Grothendieck topology S$S$ on P$P$ such that, for all x$x$ in C$\mathbf{C}$ and all y$y$ in the algebraic closure of Qp$\mathbf{Q}_p$, the sieves {q ≤ p : x ∈ dom(q)}$\{q \le p : x \in \mathrm{dom}(q)\}$ and {q ≤ p : y ∈ rng(q)}$\{q \le p : y \in \mathrm{rng}(q)\}$ are both covering sieves at p$p$. (Any larger Grothendieck topology will do; for forcing one uses the double negation topology which includes this one.) Note that the points of (the locale associated to) the site (P, S)$(P, S)$ are in one-to-one correspondence with isomorphisms between C$\mathbf{C}$ and the algebraic closure of Qp$\mathbf{Q}_p$.
Now, the isomorphisms between C$\mathbf{C}$ and the algebraic closure of Qp$\mathbf{Q}_p$ correspond precisely with geometric morphisms Set → Sh(P, S)$\mathrm{Set} \rightarrow \mathrm{Sh}(P, S)$. Whatever is preserved by this geometric morphism can be done equally well on either side. In other words, many things that can be done in Set$\mathrm{Set}$ using such an isomorphism can also be done in Sh(P, S)$\mathrm{Sh}(P, S)$ without this assumption. Of course, this heavily depends on what needs to be done, but there are known ways to carry out this kind of analysis. Since the site (P, S)$(P, S)$ is relatively nice, this analysis is far from impossible.
It's interesting to see how this formalizes Emerton's view. Objects of Sh(P, S)$\mathrm{Sh}(P, S)$ are functors F:Pop→Set$F:P^{\mathrm{op}} \rightarrow \mathrm{Set}$, subject to the usual continuity requirements. One can think of F$F$ as a set which evolves along P$P$. This makes sense since we should think of partial isomorphisms p ∈ P$p \in P$ as approximations to the desired isomorphism from C$\mathbf{C}$ onto the algebraic closure of Qp$\mathbf{Q}_p$. As more and more information is packed into p$p$, we gain more and more information about the stalk of F$F$ at the given point. Although he only considers the first few approximations in his answer, Emerton's view corresponds precisely to working in Set$\mathrm{Set}$ while keeping in mind that the work being done could be done equally well in Sh(P, S)$\mathrm{Sh}(P, S)$ instead.
