The number of complex numbers z satisfying z^3=bar z is

Take moduli: $|z|^3=|z|$, so $z=0$ or $|z|=1$. For $|z|=1$ we have $\bar z=1/z$, so the equation becomes $z^3=1/z$, i.e. $z^4=1$, which gives the four roots $\pm1,\pm i$. Each of them satisfies the original equation, because $\bar z=z^{-1}=z^3$ whenever $z^4=1$ and $|z|=1$. Adding the root $z=0$, the total number of solutions is $4+1=5$. A common slip is to forget $z=0$ (giving $4$).