Completeness of field may change under the extension of field, then what justifies $K^{nr}$ is complete?

Question

Completeness of field does not change under the extension of field ?

Let $K$ be a valuation field and $v$ be one of it's valuation. Let $K$ be complete with respect to $v$.

Then, arbitrary extension field of $K$ is also complete?

I heard this is not true in some case, but I couldn't find counterexamples. If the titled question does not hold, then, the fact that $K^{nr}$($K's$ maximal unramified extension) is complete follows from
another reason?

Thank you in advance.

Answer 1

The maximal unramified completion of $\mathbb Q_p$ is not complete. See here for a proof.

In fact, an infinite separable extension of a complete field is never complete.