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 Swiatek,
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 other functions.