We analyze the condensation phase transitions in out-of-equilibrium complex networks in a unifying framework which includes the nonlinear model and the fitness model as its appropriate limits. We show a novel phase structure which depends on both the fitness parameter and the nonlinear exponent. The occurrence of the condensation phase transitions in the dynamical evolution of the network is demonstrated by using Bianconi-Barabasi method. We find that the nonlinear and the fitness preferential attachment mechanisms play important roles in formation of an interesting phase structure.