  We perform a systematic study of the image of the Gauss map for complete
minimal surfaces in Euclidean four-space. In particular, we give a geometric
interpretation of the maximal number of exceptional values of the Gauss map of
a complete orientable minimal surface in Euclidean four-space. We also provide
optimal results for the maximal number of exceptional values of the Gauss map
of a complete minimal Lagrangian surface in the complex two-space and the
generalized Gauss map of a complete nonorientable minimal surface in Euclidean
four-space.
