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Define morphism of Q[i,w] in terms of i and w

I want to construct the field K=Q[i,w] where w=42 and define a field homomorphism such as wiw, ii. I'd like to do

K.<w,I> = NumberField([x^4-2,x^2+1])
H = End(K)
H([I*w,-I])

but it doesn't work because K is considered to be the relative field (Q[i])[w]. I know I could work with the absolute field

K_abs.<theta> = K.absolute_field()

but I'd really like to define the morphism in terms of w,i. Is it possible?