  We investigate high scale boundary conditions on the quartic Higgs-couplings
and their $\beta$-functions in the Type-II Two Higgs Doublet Model and the
Inert Doublet Model. These conditions are associated with two possible UV
physics scenarios: the Multiple Point Principle, in which the potential
exhibits a second minimum at $M_{Pl}$, and Asymptotic Safety, where the scalar
couplings run towards an interacting UV fixed point at high scales. We employ
renormalisation group running at two-loops and apply theoretical and
experimental constraints to their parameter spaces. We find neither model can
simultaneously accommodate the MPP whilst also providing realistic masses for
both the Higgs and the top quark. However, we do find regions of parameter
space compatible with Asymptotic Safety.
