Experiments at the LHC have been improving the measurements of the Higgs boson properties, and searches for new physics are being actively conducted. In the absence of deviations from the Standard Model thus far, it would be prudent to seek for other complementary strategies in the experiments at the energy frontier. For this purpose, we propose to study the Higgs couplings at high energy scales, relevant to addressing the naturalness problem . We focus on the energy scale-dependence of the off-shell Higgs propagation, and of the top quark Yukawa coupling. We first present the traditional RGE evolution and show the possible large modifications with the existence of extra dimensions. We discuss threshold effects in the Higgs propagator due to the existence of new states, such as a gauge singlet scalar, or possible continuum states in a nearly conformal sector. We finally illustrate the composite Higgs scenario by parameterizing the Higgs interactions by non-local form factors. We propose to exploit the unique signal process gg —> h* —> ZZ*. We find that certain scenarios are potentially observable at the LHC upgrade to a high luminosity or higher energy.