In my first post, I asked if $\mathbb{Q}_p$ is the completion of $\mathbb{Q}$ over some valuation of an prime ideal (and it seems to be true, according to an answer). Now I am asking if this can be generalized for finite extensions of $\mathbb{Q}_p$.
To put it in other words: Let $K$ be a finite extension of $\mathbb{Q}_p$. Question: Is there a valuation $v$ of some prime ideal in $\mathcal{O}_k$ on a global field $k$ (i.e. a finite extension of $\mathbb{Q}$) such that the completion $k_v$ of $k$ over $v$ is equal to $K$?
Yes. Let $K$ be a finite extension of $\mathbb Q_p$. We want to show that there is a finite extension $F$ of $\mathbb Q$, and a prime ideal $\mathfrak p$ in the ring of integers $\mathcal O_F$ of $F$, such that $K$ is the completion of $F$ with respect to the $\mathfrak p$-adic valuation on $F$.
Krasner's lemma tells us that there is an irreducible polynomial $g(t)$ (irreducible in the ring $\mathbb Q_p[t]$, that is) with rational coefficients, and a root $\beta$ of $g(t)$, such that $K = \mathbb Q_p(\beta)$. Then $g(t)$ is also irreducible in the ring $\mathbb Q[t]$, so if we set $F = \mathbb Q(\beta)$, then $[F : \mathbb Q] = [K : \mathbb Q_p] = \operatorname{deg} g$.
Now let $| - |$ be the unique extension of the $p$-adic absolute value to $K$. Restrict $|-|$ to an absolute value on $F$. A version of Ostrowski's theorem for number fields states that this absolute value is either archimedean, or arises from a prime ideal $\mathfrak p$ in the ring of integers $\mathcal O_F$ of $F$. Specifically, there is a real number $0 < \rho < 1$ and a prime ideal $\mathfrak p$ such that
$$|x| = \rho^{\operatorname{ord}_{\mathfrak p}(x)}.$$
Obviously this absolute value is not archimedean on $F$ (its restriction to the $p$-adic absolute value on $\mathbb Q$ isn't, after all). Anyway, the completion of $F$ with respect to $\mathfrak p$ is the same as the metric space completion of $F$ with respect to $|-|$. Finally, just as the metric space completion of $\mathbb Q$ with respect to $|-|$ gets you $\mathbb Q_p$, so does the metric space completion of $F = \mathbb Q(\beta)$ with respect to $|-|$ get you $\mathbb Q_p(\beta) = K$.