Function $z\mapsto\overline{P(z)}$ has at most $3d-2$ fixed points, where $d\geq 2$ is the degree of $P$, and this is best possible. This remarkable result is due to Khavinson and Świa̧tek,
MR1933331 Khavinson, Dmitry, Świa̧tek, Grzegorz , On the number of zeros of certain harmonic polynomials, Proc. Amer. Math. Soc. 131 (2003), no. 2, 409–414.
and it was later generalized to rational functions, and to some transcendental functions. There is a survey of related results:
D. Khavinson and G. Neumann, From the fundamental theorem of algebra to astrophysics: a “harmonious” path, Notices Amer. Math. Soc. 55 (2008), no. 6, 666–675.
Let me mention a major unsolved problem: let $p$, $q$ be polynomials of degrees $m>n$. How many solutions can the equation $$\overline{p(z)}=q(z)$$ have? Can one do better than the Bézout estimate $mn$?
