Fisher's geometric model was originally introduced to argue that complex adaptations must occur in small steps because of pleiotropic constraints. When supplemented with the assumption of additivity of mutational effects on phenotypic traits, it provides a simple mechanism for the emergence of genotypic epistasis from the nonlinear mapping of phenotypes to fitness. Of particular interest is the occurrence of sign epistasis, which is a necessary condition for multipeaked genotypic fitness landscapes. Here we compute the probability that a pair of randomly chosen mutations interacts sign-epistatically, which is found to decrease algebraically with increasing phenotypic dimension n, and varies non-monotonically with the distance from the phenotypic optimum. We then derive asymptotic expressions for the mean number of fitness maxima in genotypic landscapes composed of all combinations of L random mutations. This number increases exponentially with L, and the corresponding growth rate is used as a measure of the complexity of the genotypic landscape. The dependence of the complexity on the parameters of the model is found to be surprisingly rich, and three distinct phases characterized by different landscape structures are identified. The complexity generally decreases with increasing phenotypic dimension, but a non-monotonic dependence on n is found in certain regimes. Our results inform the interpretation of experiments where the parameters of Fisher's model have been inferred from data, and help to elucidate which features of empirical fitness landscapes can (or cannot) be described by this model.