**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 $\mathbb{C}\mathbb{P}^1$.



Classification of complex structures on $\mathbb{R}^{2n}$
There exist *infinitely many* inequivalent complex structures in $\mathbb{R}^{2n}$ for all $n \geq 2$.


See for instance the paper 


K. Diederich, N. Sibony: *[Strange complex structures on Euclidean space](https://eudml.org/doc/152188)*, Journal für die reine und angewandte Mathematik **311-312**, page 397-407 (1979)


and the references given therein.


