  In this paper, we show that if $(X,g)$ is an oriented four dimensional
Einstein manifold which is self-dual or anti-self-dual then superminimal
surfaces in $X$ of appropriate spin enjoy the Calabi-Yau property, meaning that
every immersed surface of this type from a bordered Riemann surface can be
uniformly approximated by complete superminimal surfaces with Jordan
boundaries. The proof uses the theory of twistor spaces and the Calabi-Yau
property of holomorphic Legendrian curves in complex contact manifolds.
