We obtain a closed formula for the Kähler potential of a broad class of four-dimensional Lorentzian or Euclidean conformal "Kähler" geometries, including the Plebański-Demiański class and various gravitational instantons such as Fubini-Study and Chen-Teo. We show that the Kähler potentials of Schwarzschild and Kerr are related by a Newman-Janis shift. Our method also shows that a class of supergravity black holes, including the Kerr-Sen spacetime, is Hermitian. We finally show that the integrability conditions of complex structures lead naturally to the Weyl double copy.