Proposals of measuring the off-shell Higgs contributions and first measurements at the LHC have electrified the Higgs phenomenology community for two reasons: Firstly, probing interactions at high invariant masses and momentum transfers is intrinsically sensitive to new physics beyond the Standard Model, irrespective of a resonant or non-resonant character of a particular BSM scenario. Secondly, under specific assumptions a class of models exists for which the off-shell coupling measurement together with a measurement of the on-shell signal strength can be re-interpreted in terms of a bound on the total Higgs boson width.  In this paper, we provide a first step towards a classification of the models for which a total width measurement is viable and we discuss examples of BSM models for which the off-shell coupling measurement can be important in either constraining or even discovering new physics in the upcoming LHC runs. Specifically, we discuss the quantitative impact of the  presence of dimension six operators on the (de)correlation of Higgs on- and off-shell regions keeping track of all interference effects. We furthermore investigate off-shell measurements in a wider context of new (non-)resonant physics in Higgs portal scenarios and the MSSM.