  The resonance-continuum interference is usually neglected when the width of a
resonance is small compared to the resonance mass. We re-examine this standard
by studying the interference effects in high-resolution decay channels,
$\gamma\gamma$ and $ZZ$, of the heavy Higgs boson $H^0$ in nearly aligned
two-Higgs-doublet models. For the $H^0$ with a sub-percent width-to-mass ratio,
we find that, in the parameter space where the LHC 14 TeV $ZZ$ resonance search
can be sensitive, the interference effects can modify the $ZZ$ signal rate by
${\cal O}(10)\%$ and the exclusion reach by ${\cal O}(10)$ GeV. In other
parameter space where the $ZZ$ or $\gamma\gamma$ signal rate is smaller, the
LHC 14 TeV reach is absent, but a resonance shape can be much more dramatically
changed. In particular, the $\gamma\gamma$ signal rate can change by ${\cal
O}(100)\%$. Relevant to such parameter space, we suggest variables that can
characterize a general resonance shape. We also illustrate the relevance of the
width on the interference by adding non-standard decay modes of the heavy Higgs
boson.
