In the mathematical modelling of real-life applications, systems of equations with complex coefficients often arise. While most techniques of numerical linear algebra, e.g., Krylov-subspace methods, extend directly to the case of complex-valued matrices, many of the most effective preconditioning techniques and linear solvers are limited to the real-valued case. In this talk, we present the extension of the algebraic multigrid method to such complex-valued systems. Algebraic multigrid (AMG) solvers and preconditioners are often the methods of choice for linear systems that arise from the discretization of elliptic or parabolic PDEs. While much of the research into AMG is motivated by applications such as fluid flow and solid mechanics, we consider the design of optimal algebraic multigrid solvers for problems such as those arising from models of electromagnetics and quantum mechanics. The complex multigrid components are motivated by a combination of classical multigrid considerations and experiments with local Fourier analysis. We present results for the complex-shift Helmholtz equation on unstructured grids and for random complex matrices related to lattice gauge theory.