If the diphoton event is indeed pointing to a bonafide new particle (and I am a little skeptical at this point), and we make the additional assumption that the current exclusions on other searches hold, then I think this is pointing to rather complicated new physics, at least if we insist on naturalness and other niceties of model building.
In particular its very hard make the MSSM (in the decoupling limit) work with this data. The existence of this particle would very likely (depending on how the measurement of the width holds up) nearly completely exclude most of its parameter space, and you would need to take extreme limits in order not to damage the existing known standard model like couplings for the regular Higgs. Likely one needs an extension of the MSSM to something like say the nMSSM to have enough model building wriggle room. Similarly its been pointed out to me that it is also difficult to make a Higgs doublet model work for very similar reasons.
Anyway, this is nonminimal enough that it's going to occupy the majority of theorists time for the next year while the experimental situation is worked out.