We introduce two novel generalizations of the Koopman operator method of nonlinear dynamic modeling. Each of these generalizations leads to greatly improved predictive performance without sacrificing a unique trait of Koopman methods: the potential for fast, globally optimal control of nonlinear, nonconvex systems. The first generalization, Convex Koopman Models, uses convex rather than linear dynamics in the lifted space. The second, Extended Koopman Models, additionally introduces an invertible transformation of the control signal which contributes to the lifted convex dynamics. We describe a deep learning architecture for parameterizing these classes of models, and show experimentally that each significantly outperforms traditional Koopman models in trajectory prediction for two nonlinear, nonconvex dynamic systems.