  We study $R^2$-Higgs inflation in a model with two Higgs doublets in which
the Higgs sector of the Standard Model is extended by an additional Higgs
doublet, thereby four scalar fields are involved in the inflationary
evolutions. We first derive the set of equations required to follow the
inflationary dynamics in this two Higgs doublet model, allowing a nonminimal
coupling between the Higgs-squared and the Ricci scalar $R$, as well as the
$R^2$ term in the covariant formalism. By numerically solving the system of
equations, we find that, in parameter space where a successful $R^2$-Higgs
inflation are realized and consistent with low energy constraints, the
inflationary dynamics can be effectively described by a single slow-roll
formalism even though four fields are involved in the model. We also argue that
the parameter space favored by $R^2$-Higgs inflation requires nearly degenerate
masses for $m_\mathsf{H}$, $m_A$ and $m_{\mathsf{H}^\pm}$, where $\mathsf{H}$,
$A$, and $\mathsf{H}^\pm$ are the extra CP even, CP odd, and charged Higgs
bosons in the general two Higgs doublet model taking renormalization group
evolutions of the parameters into account. Discovery of such heavy scalars at
the Large Hadron Collider (LHC) are possible if they are in the sub-TeV mass
range. Indirect evidences may also emerge at the LHCb and Belle-II experiments,
however, to probe the quasi degenerate mass spectra one would likely require
high luminosity LHC or future lepton colliders such as the International Linear
Collider and the Future Circular Collider.
