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We present a systematic study of an extension of the Standard Model (SM) withtwo Higgs doublets and one complex singlet (2HDM+S). In order to gainanalytical understanding of the parameter space, we re-parameterize the 27parameters in the Lagrangian by quantities more closely related to physicalobservables: physical masses, mixing angles, trilinear and quadratic couplings,and vacuum expectation values. Embedding the 125\,GeV SM-like Higgs bosonobserved at the LHC places stringent constraints on the parameter space. Inparticular, the mixing of the SM-like interaction state with the remainingstates is severely constrained, requiring approximate alignment withoutdecoupling in the region of parameter space where the additional Higgs bosonsare light enough to be accessible at the LHC. In contrast to 2HDM models, largebranching ratios of the heavy Higgs bosons into two lighter Higgs bosons or alight Higgs and a $Z$ boson, so-called Higgs cascade decays, are ubiquitous inthe 2HDM+S. Using currently available limits, future projections, and our owncollider simulations, we show that combining different final states arisingfrom Higgs cascades would allow to probe most of the interesting region ofparameter space with Higgs boson masses up to 1\,TeV at the LHC with$L=3000\,{\rm fb}^{-1}$ of data.

Two Higgs Doublets and a Complex Singlet: Disentangling the Decay Topologies and Associated Phenomenology