The evolution of complex organisms is a puzzle for evolutionary theory because beneficial mutations should be less frequent in complex organisms, an effect termed "cost of complexity." However, little is known about how the distribution of mutation fitness effects (f(s)) varies across genomes. The main theoretical framework to address this issue is Fisher's geometric model and related phenotypic landscape models. However, it suffers from several restrictive assumptions. In this paper, we intend to show how several of these limitations may be overcome. We then propose a model of f(s) that extends Fisher's model to account for arbitrary mutational and selective interactions among n traits. We show that these interactions result in f(s) that would be predicted by a much smaller number of independent traits. We test our predictions by comparing empirical f(s) across species of various gene numbers as a surrogate to complexity. This survey reveals, as predicted, that mutations tend to be more deleterious, less variable, and less skewed in higher organisms. However, only limited difference in the shape of f(s) is observed from Escherichia coli to nematodes or fruit flies, a pattern consistent with a model of random phenotypic interactions across many traits. Overall, these results suggest that there may be a cost to phenotypic complexity although much weaker than previously suggested by earlier theoretical works. More generally, the model seems to qualitatively capture and possibly explain the variation of f(s) from lower to higher organisms, which opens a large array of potential applications in evolutionary genetics.