We study the off-shell production of the Higgs boson at the LHC to probe Higgs physics at higher energy scales utilizing the process gg → h à → ZZ. We focus on the energy scale dependence of the off-shell Higgs propagation, and of the top quark Yukawa coupling, y t ðQ 2 Þ. Extending our recent study [D. Goncalves, T. Han, and S. Mukhopadhyay, Phys. Rev. Lett. 120, 111801 (2018).], 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 ðQ 2 Þ 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 nonlocal 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 the 2σ level at the high-luminosity 14 TeV HL-LHC for a new physics scale up to Oð1 TeVÞ, and at the upgraded 27 TeV HE-LHC for a scale up to Oð3 TeVÞ. For a few favorable scenarios, 5σ level observation may be possible at the HE-LHC for a scale of about Oð1 TeVÞ.