  Given a perfectoid field, we find an elementary extension and a henselian
defectless valuation on it, whose value group is divisible and whose residue
field is an elementary extension of the tilt. This specializes to the almost
purity theorem over perfectoid valuation rings and Fontaine-Wintenberger. Along
the way, we prove an Ax-Kochen/Ershov principle for certain deeply ramified
fields, which also uncovers some new model-theoretic phenomena in positive
characteristic. Notably, we get that the perfect hull of $\mathbb{F}_p(t)^h$ is
an elementary substructure of the perfect hull of $\mathbb{F}_p(\!(t)\!)$.
