  This paper focuses on the study of three dimensional real hypersurfaces in
non-flat complex space forms whose $^{*}$-Ricci tensor satisfies conditions of
parallelism. More precisely, extension of existing results concerning real
hypersurfaces with vanishing, semi-parallel and pseudo-parallel $^{*}$-Ricci
tensor in case of ambient space being the complex hyperbolic space are
provided. Furthermore, new results concerning $\xi$-parallelism of $^{*}$-Ricci
tensor of real hypersurfaces in non-flat complex space forms are presented.
