This question is somewhat silly; I'm a bit confused and asking for clarification. A symplectic manifold is a (real) manifold equipped with a closed non-degenerate 2-form, or equivalently an integrable $\text{Sp}(2n, \mathbb{R})$-structure. The group $\text{Sp}(2n, \mathbb{R})$ is defined as the group of linear automorphisms of $\mathbb{R}^n \oplus \mathbb{R}^n$ which preserve the standard symplectic form
$$ \omega = dx_1 \wedge dy_1 + \cdots + dx_n \wedge dy_n $$
There also exists the group $\text{Sp}(2n, \mathbb{C})$, and that's the one I'm a bit confused about. As far as I can see, it seems to be again defined as the group of linear automorphisms of $\mathbb{C}^n \oplus \mathbb{C}^n$ preserving a form defined by the same formula as above, but apparently it might not?
Furthermore, I'm curious as to the geometrical significance of this complex symplectic group. For example, do people study manifolds equipped with integrable $\text{Sp}(2n, \mathbb{C})$-structures? Those would need to have real dimension $4n$, I suppose. My first instinct was to guess such a thing would be a symplectic structure plus a compatible complex structure -- but there's no reason for such things to require real dimension a multiple of $4$, is there? I don't really understand what's going on here, as I think I've never heard of using this group to do symplectic geometry.