Axion-like particles (ALPs) are promising candidates for light dark matter (DM), and their dynamics is described by the dark sector of Lagrangian. It is usually assumed that the dark sector is CP conserving. However, it does not have to be so. Actually, CP is violating in the weak interaction in the standard model. In this talk, we introduce a new renormalizable model with CP violation in the dark sector and discuss the properties and testability of the predicted ALP. In addition, we discuss a phenomenon of Higgs bosons which is happening in a specific situation, focusing on this new ALP model as a concrete model. Namely, the mixing between two Higgs states is suppressed by quantum effects in case that the extra Higgs boson has large decay width. This might be a new decoupling regime of extra Higgs bosons. This talk is based on arXiv:2111.03653 and arXiv:2204.01739.