  We study the off-shell production of the Higgs boson at the LHC to probe
Higgs physics at higher energy scales utilizing the process $g g \rightarrow
h^{*} \rightarrow ZZ$. We focus on the energy scale dependence of the off-shell
Higgs propagation, and of the top quark Yukawa coupling, $y_t (Q^2)$. Extending
our recent study in arXiv:1710.02149, we first discuss threshold effects in the
Higgs propagator due to the existence of new states, such as a gauge singlet
scalar portal, and a possible continuum of states in a conformal limit, both of
which would be difficult to discover in other traditional searches. We then
examine the modification of $y_t (Q^2)$ from its Standard Model (SM) prediction
in terms of the renormalization group running of the top Yukawa, which could be
significant in the presence of large flat extra-dimensions. Finally, we explore
possible strongly coupled new physics in the top-Higgs sector that can lead to
the appearance of a non-local $Q^2$-dependent form factor in the effective
top-Higgs vertex. We find that considerable deviations compared to the SM
prediction in the invariant mass distribution of the $Z$-boson pair can be
conceivable, and may be probed at a $2\sigma$-level at the high-luminosity 14
TeV HL-LHC for a new physics scale up to $\mathcal{O}(1 {~\rm TeV})$, and at
the upgraded 27 TeV HE-LHC for a scale up to $\mathcal{O}(3 {~\rm TeV})$. For a
few favorable scenarios, $5\sigma$-level observation may be possible at the
HE-LHC for a scale of about $\mathcal{O}(1 {~\rm TeV})$.
