Monomial functions, normal polynomials and polynomial equations

In this paper we consider generalized monomial functions $f, g\colon \mathbb{F}\to \mathbb{C}$ (of possibly different degree) that also fulfill $$f(P(x))= Q(g(x)) \qquad \left(x\in \mathbb{F}\right),$$ where $P\in \mathbb{F}[x]$ and $Q\in \mathbb{C}[x]$ are given (classical) polynomials.