  Experimental data on rare $B$-meson decays indicate deviations from Standard
Model predictions. In studies of these decays, possible new sources of CP
violation are often neglected. We discuss CP violation in the rare $B$-meson
decays $B\to K\ell^+\ell^- (\ell = \mu,e)$ and point to two phenomena that
arise when new sources of CP violation are included. First, the Wilson
coefficients $C_{9\ell}$ and $C_{10\ell}$ become complex, and we show how we
can extract their values from measurements of direct and mixing-induced CP
asymmetries. Second, new sources of CP violation can generate nontrivial lepton
flavour universality violation. Such a violation is usually measured through
ratios like $R_K$ and $R_{K^*}$, but we show that measuring only these ratios
leaves a large parameter space unexplored. These results bring exciting
opportunities to reveal New Physics effects in the high-precision era.
