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When you construct the tower of extensions explicitly, you can do the following:

sage: K.<I> = NumberField(x^2+1)
sage: L.<w> = K.extension(x^4-2)
sage: f = L.hom([I*w], base_map=K.hom([-I])); f
Relative number field endomorphism of Number Field in w with defining polynomial x^4 - 2 over its base field
  Defn: w |--> I*w
        I |--> -I
sage: f(I + w)
I*w - I