First, let me observe that it is consistent with $\mathsf{ZF}$ + $\mathsf{DC}$ 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 $\mathbf{C}$ or $\mathbf{Q}_p$. 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 $P$ below is countably closed and ignore the rest of this post.)
Let $P$ be the poset of field isomorphisms $p:A\rightarrow B$ where $A$ is a countable subfield of $\mathbf{C}$ and $B$ is a countable subfield of the algebraic closure of $\mathbf{Q}_p$, and $p \le q$ iff $p \supseteq q$ (i.e. $q$ is a restriction of $p$). This ordering is slightly counterintuitive, but it is more convenient than the opposite. The poset $P$ can be viewed as a category where there is one and only one arrow between any two objects $p$ and $q$ iff $p \le q$. The poset $P$ then becomes a Cartesian category where the terminal object is the isomorphism between the two copies of $\mathbf{Q}$ in each field, and the product of $p$ and $q$ are is the intersection of the (graphs of) $p$ and $q$.
There are many Grothendieck topologies that one could define on $P$. The relevant one for our context is the smallest Grothendieck topology $S$ on $P$ such that, for all $x$ in $\mathbf{C}$ and all $y$ in the algebraic closure of $\mathbf{Q}_p$, the sieves $\{q \le p : x \in \mathrm{dom}(q)\}$ and $\{q \le p : y \in \mathrm{rng}(q)\}$ are both covering sieves at $p$. (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 $(P, S)$ are in one-to-one correspondence with isomorphisms between $\mathbf{C}$ and the algebraic closure of $\mathbf{Q}_p$.
Now, the isomorphisms between $\mathbf{C}$ and the algebraic closure of $\mathbf{Q}_p$ correspond precisely with geometric morphisms $\mathrm{Set} \rightarrow \mathrm{Sh}(P, S)$. 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 $\mathrm{Set}$ using such an isomorphism can also be done in $\mathrm{Sh}(P, S)$ 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 $(P, S)$ is relatively nice, this analysis is far from impossible.
It's interesting to see how this formalizes Emerton's view. Objects of $\mathrm{Sh}(P, S)$ are functors $F:P^{\mathrm{op}} \rightarrow \mathrm{Set}$, subject to the usual continuity requirements. One can think of $F$ as a set which evolves along $P$. This makes sense since we should think of partial isomorphisms $p \in P$ as approximations to the desired isomorphism from $\mathbf{C}$ onto the algebraic closure of $\mathbf{Q}_p$. As more and more information is packed into $p$, we gain more and more information about the stalk of $F$ at the given point. Although he only considers the first few approximations in his answer, Emerton's view corresponds precisely to working in $\mathrm{Set}$ while keeping in mind that the work being done could be done equally well in $\mathrm{Sh}(P, S)$ instead.
