First thing don't worry about field equality as this leads to unnecessarily abstract concepts. Both fields are subsets of complex numbers so you can treat this just as a question about equality of subsets. Sou you just prove two inclusions.
I think that for the complete proof you need these two observations:
1 Multiplication by unitary matrix does not change the norm of a vector
see $(Ux)^*(Ux)=x^*U^*Ux=x^*Ix=x^*x$ where $I$ is the identity matrix.
2 Unitary matrix is regular. In other words, it is surjective as a linear operator on a vector space.
Now the equation $x^*U^*AUx=(Ux)^*A(Ux)$ will allow you to complete the proof. Just for each value in the one field of values find a vector that gives you this value and try to prove the existence of a vector that gives you the same value in the other field of values.