  It is possible to ameliorate the Higgs vacuum stability problem by switching
over to two Higgs doublet models (2HDM), ensuring a stable electroweak vacuum
up to the Planck scale, even though the top quark mass may be on the high side.
However, the simultaneous requirements of perturbative unitarity, and also
compatibility with collider and flavour data, constrain the parameter space
severely. We investigate the collider signals answering to the regions allowed
by such constraints. In particular, the near degeneracy of the neutral heavy
scalar and the pseudoscalar is a feature that is probed. The LHC allows
distinguishability of these two states, together with signal significance of at
least 3$\sigma$, in its high-luminosity run. While $e^+ e^-$ colliders may have
rather low event rates, muon colliders, cashing on the principle of radiative
return, can probe 2HDM scenarios with (pseudo)scalar masses up to a TeV or so,
though with the price of losing distinction between the CP-even and odd states.
