  On a complex manifold $(M,J)$, we interpret complex symplectic and
pseudo-K\"ahler structures as symplectic forms with respect to which $J$ is,
respectively, symmetric and skew-symmetric. We classify complex symplectic
structures on 4-dimensional Lie algebras. We develop a method for constructing
hypersymplectic structures from the above data. This allows us to obtain an
example of a hypersymplectic structure on a 4-step nilmanifold.
