We investigate the conjecture that the horizon of an index-fractional Brownian surface has (almost surely) the same H older exponents as the surface itself, with corresponding relationships for fractal dimensions. We establish this formally for the usual Brownian surface (where = 1 2 ), and also for other ; 0 < < 1, assuming a hypothesis concerning maxima of index-Brownian motion. We provide computational evidence that the conjecture is indeed true for all .