  When building CP-symmetric models beyond the Standard Model, one can impose
CP-symmetry of higher order. This means that one needs to apply the
CP-transformation more than two times to get the identity transformation, but
still the model is perfectly CP-conserving. A multi-Higgs-doublet model based
on CP-symmetry of order 4, dubbed CP4, was recently proposed and its
phenomenology is being explored. Here, we show that the construction does not
stop at CP4. We build examples of renormalizable multi-Higgs-doublet potentials
which are symmetric under CP8 or CP16, without leading to any accidental
symmetry. If the vacuum conserves CP-symmetry of order 2k, then the neutral
scalars become CP-eigenstates, which are characterized not by CP-parities but
by CP-charges defined modulo 2k. One or more lightest states can be the dark
matter candidates, which are protected against decay not by the internal
symmetry but by the exotic CP. We briefly discuss their mass spectra and
interaction patterns for CP8 and CP16.
