  Four dimensional simply connected Lie groups admitting a pseudo K\"ahler
metric are determined. The corresponding Lie algebras are modelized and the
compatible pairs $(J,\omega)$ are parametrized up to complex isomorphism (where
$J$ is a complex structure and $\omega$ is a symplectic structure). Such
structure gives rise to a pseudo Riemannian metric $g$ for which $J$ is
parallel. It is proved that most of these complex homogeneous spaces admit a
pseudo K\"ahler Einstein metric. Ricci flat and flat metrics are determined. In
particular Ricci flat unimodular K\"ahler Lie algebras are flat in dimension
four. Other algebraic and geometric features are treated. A general
construction of Ricci flat pseudo K\"ahler structures in higher dimensions on
some affine Lie algebras is given. Walker and hypersymplectic metrics on Lie
algebras are compared.
