Q1. Let $K$ be a local field with valuation $v$. Let us call $K'\subset K$ a nice local subfield if it is complete with respect to the induced from $K$ valuation. By local subfield I will mean a subfield $K'\subset K$ which is complete with respect to some valuation $v'$, possibly different from the one induced from $K$. Is it true that any local subfield is a nice local subfield? In particular, I'm interested in the case when $K$ is a higher local field.

Q2. Is it true that any local subfield of a higher local field is a higher local field? Is it true in the case of nice local fields?