Suppose $f:\mathbb{C}\to\mathbb{C}$ is a map with your favorite smoothness condition (say, $C^1, C^{\infty}$ or holomorphic) and suppose that $f(\overline{\mathbb{Q}})\subset\overline{\mathbb{Q}}$. Is $f$ a polynomial? (I.e., if you chose $C^1$ or $C^{\infty}$ in the beginning, then is $f$ a polynomial in $z$ and $\overline{z}$, and if you chose holomorphic, then is $f$ a polynomial in $z$?)

I imagine this is a well-known problem, but I don't remember having seen it before. A holomorphic counter-example would be quite impressive, though I doubt such a thing exists.