Models beyond the Standard Model equipped with extended Higgs sectors demonstrate a surprising variety of ways how the CP symmetry can be imposed and violated. The mere definition of the CP transformation is not uniquely fixed a priori, and it happens that in certain multi-Higgs models CP symmetry can take exotic form. I will present the first ever model which is based on CP symmetry of order 4 (CP4), which means that one must apply it four time to produce identity transformation. Imposing CP4 without accidental symmetries requires at least three Higgs doublets and leads to a very characteristic scalar sector. If CP4 remains unbroken, it leads to scalar dark matter candidates stabilized by CP symmetry, and can also generate neutrino masses with naturally mild hierarchy. If spontaneously broken, CP4 can be extended to the Yukawa sector and leads characteristic flavor physics prediction. In short, CP4 3HDM emerges as a novel and rich framework whose phenomenological potential awaits exploration.
