Recall that: a symplectic form is a non-degenerate skew-symmetric $\mathbb R$-bilinear form.
The only $\mathbb R$-valued symplectic form on $\mathbb C$, $f:\mathbb C\times \mathbb C\rightarrow \mathbb R$, is $f(z,w)=\Im m (z\overline{w})$.
The only $\mathbb C$-valued symplectic form on $\mathbb C$, $f:\mathbb C\times \mathbb C\rightarrow \mathbb C$, is $f(z,w)=\alpha \, z\overline{w}; \, \alpha\in \mathbb R$.
I would like know, there is a characterization of all $\mathbb C$-valued symplectic forms on $\mathbb C^2$, $f:\mathbb C^2\times \mathbb C^2\rightarrow \mathbb C$ ?
Thank you in advance