The Kuramoto model and its generalizations have been broadly employed to characterize and mechanistically understand various collective dynamical phenomena, especially the emergence of synchrony among coupled oscillators. Despite almost five decades of research, many questions remain open, in particular, for finite-size systems. Here, we generalize recent work [Thümler et al., Phys. Rev. Lett. 130, 187201 (2023)] on the finite-size Kuramoto model with its state variables analytically continued to the complex domain and also complexify its system parameters. Intriguingly, systems of two units with purely imaginary coupling do not actively synchronize even for arbitrarily large magnitudes of the coupling strengths, |K|→∞, but exhibit conservative dynamics with asynchronous rotations or librations for all |K|. For generic complex coupling, both traditional phase-locked states and asynchronous states generalize to complex locked states, fixed points off the real subspace that exist even for arbitrarily weak coupling. We analyze a new collective mode of rotations exhibiting finite, yet arbitrarily large rotation numbers. Numerical simulations for large networks indicate a novel form of discontinuous phase transition. We close by pointing to a range of exciting questions for future research.