Multi-Higgs models. Perspectives for identification of wide set of models in future experiments at colliders in the SM-like situation

Higgs mechanism of EWSB can be realized in both well known minimal model and with more complex non-minimal Higgs models. These non-minimal models contain new Higgs bosons -- neutral $h_a$ and charged $H_b^\pm$. Necessary step in the discovery of such model is observation of these additional Higgses. We discuss the potential of such researches at modern and future colliders in the light of recent LHC results, for wide set of models, including 2HDM as a simplest example. Our conclusion is rather pessimistic. Discovery of new neutral Higgs boson at LHC is a very difficult task. Some windows to find them appear at accidentally favorable parameters of the theory. The regular way in the detection of these models is the study of processes with production of charged Higgs bosons (better, at Linear Collider).