  For conformally K\"ahler Riemannian four-manifolds with a Killing field, we
develop a framework to solve the field equations for generalised gravitational
instantons corresponding to conformal self-duality and to cosmological
Einstein-Maxwell. We obtain generic identities for the curvature of such
manifolds without assuming field equations. After applying the framework to
recover standard solutions, we find conformally self-dual generalisations of
the Page-Pope, Plebanski-Demianski, and Chen-Teo solutions, which are neither
hyper-K\"ahler nor quaternionic-K\"ahler, giving new self-dual gravitational
instantons in conformal gravity.
