  Due to the absence of tantalising hints for new physics during the LHC's run
1, the extension of the Higgs sector by dimension six operators will provide
the new phenomenological standard for searches of non-resonant extensions of
the Standard Model. Using all dominant and subdominant Higgs production
mechanisms at the LHC, we compute the constraints on Higgs physics-relevant
dimension six operators in a global and correlated fit. We show in how far
these constraints can be improved by new Higgs channels becoming accessible at
higher energy and luminosity, both through inclusive cross sections as well as
through highly sensitive differential distributions. This allows us to discuss
the sensitivity to new effects in the Higgs sector that can be reached at the
LHC if direct hints for physics beyond the SM remain elusive and the impact of
these constraints on well-motivated BSM scenarios.
