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We illustrate and discuss the phenomenology of a model featuring a two-Higgsdoublet sector augmented by two $SU(2)$ singlet scalars. The gauge symmetrygroup is extended as well with a $U(1)_{B_{i}-L_{i}}$ component whosespontaneous breaking leads to the gauge boson which has an important effect inthe muon (g-2). A global PQ symmetry is introduced upon its breaking we havethe axion particle which is also DM in our work. In particular, we havefocussed on Type-X and Type-II 2HDM models and found out that (g-2) can not beexplained only by the scalar sector for Type-II 2HDM mainly due to stringentconstraint from $b \rightarrow s \gamma$ resulted in $M_{H^{\pm}} > 800$ GeV.For Type-II 2HDM, we can have axion coupling with the gluons which generatesthe axion potential and possible explanation for the strong CP problem. Theproposed model accommodates neutrino masses via the Type-I see-saw mechanismwith an upper bound on the right-handed neutrino mass 1 GeV (1 TeV) for Type-II(Type-X) 2HDM due to the presence of Planck scale suppressed operators.Moreover, we also have additional scalars which affect the oblique parametersand hence the W-boson mass which leads us to explain the W-boson mass observedat CDF-II detector. The most stringent constraints on the masses and quarticcouplings come from the perturbativity and potential bound from belowconditions which leads to fine-tuning among the parameters in part of theparameter space. Finally, we discuss the possible detection prospects of theaxion DM and the additional gauge boson.

Axion Dark Matter and additional BSM aspects in an extended 2HDM setup