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. The complex multigrid components are motivated by a combination of classical multigrid considerations and experiments with local Fourier analysis. We present results for the algorithm applied to linear systems arising from time-harmonic reduction of the Maxwell equations and for random matrices related to lattice gauge theory.