  In this paper, we consider the anisotropic $\alpha$-Gauss curvature flow for
complete noncompact convex hypersurfaces in the Euclidean space with the
anisotropy determined by a smooth closed uniformly convex Wulff shape. We show
that for all positive power $\alpha>0$, if the initial hypersurface is complete
noncompact and locally uniformly convex, then the solution of the flow exists
for all positive time.
