The ultraviolet (UV) polarization spectrum of nearby active galactic nuclei (AGN) is poorly known. The Wisconsin Ultraviolet Photo-Polarimeter Experiment and a handful of instruments on board the Hubble Space Telescope were able to probe the near- and mid-UV polarization of nearby AGN, but the far-UV band (from 1200 angs down to the Lyman limit at 912 angs) remains completely uncharted. In addition, the linewidth resolution of previous observations was at best 1.89 angs. Such a resolution is not sufficient to probe in detail quantum mechanical effects, synchrotron and cyclotron processes, scattering by electrons and dust grains, and dichroic extinction by asymmetric dust grains. Exploring those physical processes would require a new, high-resolution, broadband polarimeter with full ultraviolet-band coverage. In this context, we discuss the AGN science case for POLLUX, a high-resolution UV spectropolarimeter, proposed for the 15-meter primary mirror option of LUVOIR (a multi-wavelength space observatory concept being developed by the Goddard Space Flight Center and proposed for the 2020 Decadal Survey Concept Study).