Superization of Kaehler and hyper-Kaehler manifolds

Let $J$ be an (almost) complex structure on a manifold $\mathcal{M}$; let~ $h$ be a~pseudo-Hermitian with respect to $J$ non-degenerate symmetric form on the tangent bundle over~$\mathcal{M}$. The manifold $\mathcal{M}$ with the triple $(ω, h, J)$, where $ω$ is an almost symplectic differential 2-fo