  Null results from searches for new physics at the Large Hadron Collider (LHC)
tend to enforce the belief that new particles must be much heavier than the
weak scale. We undertake a systematic study of the interplay between Higgs
alignment and $CP$-violation in complex two-Higgs-doublet models, which enables
us to construct a $CP$-violating scenario where new Higgs bosons are close to
the weak scale after including stringent constraints from the electric dipole
moment and measurements at the LHC. In addition, we propose a novel potential
signal of $CP$-violation in the Higgs-to-Higgs decays, $h_3\to h_2 h_1$, where
$h_3, h_2$, and $h_1$ are the heaviest, second heaviest and the Standard
Model-like neutral Higgs bosons, respectively. The decay could manifest itself
in triple boson final states in $h_1h_1h_1$ and $h_1h_1Z$, which are quite
distinct and provide unique venues for new measurements at the LHC.
