**Is there anything known about classification of complex structures on $\mathbb{R}^{2n}$ up to isomorphism for $n>1$? Say, are there finitely or infinitely many isomorphism classes? If there is a reasonable moduli space, is it finite or infinite dimensional?**

Remark. For $n=1$ there exist exactly two complex structures on $\mathbb{R}^2$: one is isomorphic to $\mathbb{C}$, the other one to the unit disc. This is a special case of the uniformization theorem saying that any simply connected complex 1-dimensional manifold is isomorphic either to unit disk, or to $\mathbb{C}$, or to $\mathbb{C}\mathbb{P}^1$.