Exponential automorphisms and a problem of Mycielski

An exponential automorphism of $\mathbf{C}$ is a function $\alpha :\mathbf{C}\to \mathbf{C}$ such that $\alpha (z_1+z_2)=\alpha (z_1)+\alpha (z_2)$ and $\alpha \left(e^z\right)=e^{\alpha (z)}$ for all $z,z_1,z_2\in \mathbf{C}$. Jan Mycielski asked if $\alpha (\ln 2)=\ln 2$ and if $\alpha (2^{1\mathrm{/}k})=2^{1\mathrm{/}k}$ for $k=2,3,4$ and for all exponential automorphisms $\alpha$. These questions are answered modulo a multiple of $2\pi i$ and a root of unity.