There are various applications of complex variables in physics, ranging from electromagnetism to quantum mechanics. Whether you need a full mathematics course in Complex Analysis for physics is another matter entirely.
besides physics, we use them in electrical engineering and signal processing to describe and manipulate AC signals. i think that the class short circut took will become useful if he/she does anything reasonably sophisticated with complex variables. everybody needs to know about Euler's formula, contour integration, and Residue Theory.
the equilibrium temperature distribution in a disc, and the electrostatic potential in a charge free region are examples of functions in physics which are solutions to laplace's differential equation. such functions are called harmonic functions.
a complex analytic function is precisely a complex valued function whose real and imaginary parts are conjugate harmonic functions. thus physics and complex analysis are intimately related.
stephen hawking is famous for physical theories allowing time to assume complex values. the idea is that any analytic function at all, e.g. polynomial, trig function, implicitly defined algebraic function,.... all are illuminated by allowing them to assume complex values both in domain and range.