  Given a semi-Riemannian $4$-manifold $(M,g)$ with two distinguished vector
fields satisfying properties determined by their shear, twist and various Lie
bracket relations, a family of K\"ahler metrics $g_K$ is constructed, defined
on an open set in $M$, which coincides with $M$ in many typical examples. Under
certain conditions $g$ and $g_K$ share various properties, such as a Killing
vector field or a vector field with a geodesic flow. In some cases the K\"ahler
metrics are complete. The Ricci and scalar curvatures of $g_K$ are computed
under certain assumptions in terms of data associated to $g$. Many examples are
described, including classical spacetimes in warped products, for instance de
Sitter spacetime, as well as gravitational plane waves, metrics of Petrov type
$D$ such as Kerr and NUT metrics, and metrics for which $g_K$ is an SKR metric.
For the latter an inverse ansatz is described, constructing $g$ from the SKR
metric.
