How many complex structures are on $\mathbb{R}^{2n}$

Question

By a complex structure on $\mathbb{R}^{2n} $ I mean $\mathbb{R}$-linear $J:\mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$ such that $JJ=-I$. I asked myself how many nonisomorphic such structures are on $\mathbb{R}^{2n}$, where by isomorphic I mean that $\mathbb{R}^{2n}$'s treated as complex spaces with complex vector space structures given from $J$'s are isomorphic as complex spaces. If I would ask only for $J$'s we would seek for matrices, such that: $$A^2=-I$$ and would get infinitely many of them since there are infinitely many for the case when $n=1$ namely $$\left[\begin{matrix} a & b\\ c & -a\end{matrix}\right]$$ with $a^2+bc=-1$. But now we treat $A_1$, $A_2$ as the same structure iff there exists $B$ such that $$A_1 B=B A_2.$$

Answer 1

If $V$ is real finite dimensional vector space and $J \colon V \rightarrow V$ is a complex structure on $V$, I will write $(V, J)$ if I want to think of $V$ as a complex vector space with the structure induced by $J$. If $J_1,J_2$ are complex structures on $V$, then the complex spaces $(V,J_1)$ and $(V,J_2)$ are isomorphic because they have the same dimension. Any isomorphism of the complex vector spaces $T \colon (V,J_1) \rightarrow (V,J_2)$ will satisfy

$$ T(J_1 v) = T(iv) = iT(v) = J_2(Tv) $$

so treating $T$ as a real-linear invertible map, it will satisfy $TJ_1 = J_2 T$ which shows that any two complex structures on $V$ are conjugate.