Completeness of a pseudo-Riemannian manifold (or a manifold with affine connection, more generally) means precisely the completeness of its geodesic flow on its frame bundle. But the frame bundle of any covering space is a covering space of the frame bundle. The geodesic flow is the flow of the geodesic vector field on the frame bundle. A vector field on a manifold is complete if and only if its lift to some (hence any) covering space is complete, which is just because the flow lines both lift (as a covering space) and project.
In the Riemannian setting, Hopf-Rinow says that completeness of the flow of the geodesic vector field is precisely completeness as a metric space. But completeness as a metric space is not defined on pseudo-Riemannian manifolds; the automorphism group of the pseudo-Riemannian geometry on Minkowski space doesn't preserve any metric, as the stabilizer of point is not compact.