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][1]*, Journal für die reine und angewandte Mathematik **311-312**, page 397-407 (1979)


and the references given therein.


  [1]: https://eudml.org/doc/152188