Multivariate polynomial optimization where variables and data are complex numbers is a non-deterministic polynomial-time hard problem that arises in various applications such as electric power systems, signal processing, imaging science, automatic control, and quantum mechanics. Complex numbers are typically used to model oscillatory phenomena which are omnipresent in physical systems. We propose a complex moment/sum-of-squares hierarchy of semidefinite programs to find global solutions with reduced computational burden compared with the Lasserre hierarchy for real polynomial optimization. We apply the approach to large-scale sections of the European high-voltage electricity transmission grid. Thanks to an algorithm for exploiting sparsity, instances with several thousand variables and constraints can be solved to global optimality.