  We analyse the sensitivity of the proposed Compact Linear Collider (CLIC) to
the existence of beyond the Standard Model (SM) Higgs bosons through their
decays into pairs of massive gauge bosons $H \to VV$ and SM-like Higgses $H \to
hh$, considering CLIC centre of mass energies $\sqrt{s} = 1.4$ TeV and $3$ TeV.
We find that resonant di-Higgs searches at CLIC would allow for up to two
orders of magnitude improvement w.r.t. the sensitivity achievable by HL-LHC in
the mass range $m_H \in [250\,\mathrm{GeV},\, 1 \,\mathrm{TeV}]$. Focusing then
on a real singlet extension of the SM, we explore the prospects of heavy Higgs
searches at CLIC for probing the regions of parameter space yielding a strongly
first order electroweak phase transition that could generate the observed
matter-antimatter asymmetry of the Universe. Our study illustrates the
complementarity between CLIC and other possible future colliders like FCC-ee in
probing singlet extensions of the SM, and shows that high-energy $e^+ e^-$
colliders provide a powerful means to unravel the nature of electroweak
symmetry breaking in the early Universe.
