Simply put, complex Hadamard matrices are scaled unitary matrices with entries drawn from unit complex numbers. They appear as an essential ingredient in quantum information theory and their real members have deep connections to finite geometry and number theory. For us, in this talk, they will be the fixed points of both discrete and continuous dynamical systems. We begin by introducing complex Hadamards and some essential preliminaries. We then discuss a discrete-time dynamical system which can be used to generate complex Hadamards as well as closely related objects known as mutually unbiased bases. Finally, we construct a continuous system whose fixed points are complex Hadamards and exploit classical results from dynamical systems theory to study local structure in spaces of complex Hadamards.