Concerning the value distribution problem for generalized Gauss maps, we not only generalize Fujimoto's theorem to complete space-like stationary surfaces in 4-dimensional Minkowski space, but also estimate the upper bound of the number of exceptional values when the Gauss image lies in the graph of a rational function f of degree m, showing a sharp contrast to Bernstein type results for minimal surfaces in  4-dimensional Euclidean space. Moreover, we introduce the conception of conjugate similarity on the special linear group to classify all degenerate stationary surfaces (i.e. m=0 or 1), and establish several structure theorems for complete stationary graphs in  4-dimensional Minkowski space from the viewpoint of the degeneracy of Gauss maps.