The Compact Linear Collider (CLIC) is a mature option for a future electron-positron collider operating at center-of-mass energies of up to 3 TeV. CLIC is foreseen in a staged approach with three center-of-mass energies, currently assumed to be 380, 1.5 and 3 TeV. This contribution discusses the physics potential of CLIC in the field of Higgs physics, based on the benchmark analyses using full detector simulation. The initial stage of operation allows study of Higgs production in Higgsstrahlung and WW-fusion, resulting in precise measurements of the production cross sections and the total Higgs-boson decay width. Operation at high energies will provide high-statistics samples of Higgs bosons enabling tight constraints on Higgs couplings and measurement of the Higgs self-coupling.