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

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