  Most numerical methods for time integration use real time steps. Complex time
steps provide an additional degree of freedom, as we can select the magnitude
of the step in both the real and imaginary directions. By time stepping along
specific paths in the complex plane, integrators can gain higher orders of
accuracy or achieve expanded stability regions. We show how to derive these
paths for explicit and implicit methods, discuss computational costs and
storage benefits, and demonstrate clear advantages for complex-valued systems
like the Schrodinger equation. We also explore how complex time stepping also
allows us to break the Runge-Kutta order barrier, enabling 5th order accuracy
using only five function evaluations for real-valued differential equations.
