Complex-valued dynamics can be used for modeling rotationally invariant two-input two-output systems and bandpass systems when they are considered in the baseband. In a few instances, control design has been done in the complex domain, which facilitated analysis and synthesis. While previous work has been application specific, we will discuss more generally how complex valued dynamics arise, basic properties of these systems, revisit some classic control theoretic results in the complex setting, and discuss two novel examples of control design in the complex domain — accelerator cavity field control and feedback linearization of RF amplifiers.