Let $p \neq q$ be distinct primes. Is there a field isomorphism $\mathbb{C}_p \cong \mathbb{C}_q$? Is there a field isomorphism $\mathbb{C}_p \cong \mathbb{C}$?
If such an isomorphism exists, given $x \in \mathbb{C}_p$, does the expression $|x|_q$ makes sense?
Yes, $\mathbb{C}_p\cong\mathbb{C}_q\cong\mathbb{C}$. However, the isomorphism is only an isomorphism of fields. The natural topology on the three fields are different, so they are nonisomorphic as topological fields.
However, the isomorphism is impossible to fully specify using a finite number of symbols/words.
Yes, for $x\in\mathbb{C}_p$, it does make sense to consider $|x|_q$. However, there is no canonical isomorphism, and thus $|x|_q$ would heavily depend on the isomorphism you choose. In fact, there are even lots of automorphisms of $\mathbb{C}$. For example there is an automorphism which sends $e$ to $\pi$, or $\pi$ to $\pi+1$.
There are a lot of easy to find resources on this. I'd suggest you google "algebraically closed field of characteristic 0".
Also you could read: https://mathoverflow.net/questions/25344/uncountable-algebraically-closed-field-other-than-c