Let $(K,\nu)$ be a field with (non-discrete) valuation. Is it possible to have a situation where $(F,\omega)$ is an immediate quadratic extension of this field with valuation such that $(F,\omega)$ is a maximally complete field?
Some things I already know:
- the residue characteristic has to be two,
- in certain specific cases this cannot occur by results of Artin-Schreier and Kedlaya-Poonen.