A search is presented for a high-mass Higgs boson in the H → ZZ → ℓ+ℓ−ℓ+ℓ−, H → ZZ → ℓ+ℓ−ν¯ν , H → ZZ → ℓ+ℓ−q¯q, and H → ZZ → ν¯νq¯q decay modes using the ATLAS detector at the CERN Large Hadron Collider. The search uses proton–proton collision data at a centre-of-mass energy of 8 TeV corresponding to an integrated luminosity of 20.3 fb−1. The results of the search are interpreted in the scenario of a heavy Higgs boson with a width that is small compared with the experimental mass resolution. The Higgs boson mass range considered extends up to 1 TeV for all four decay modes and down to as low as 140 GeV, depending on the decay mode. No significant excess of events over the StandardModel prediction is found. A simultaneous fit to the four decay modes yields upper limits on the production cross-section of a heavy Higgs boson times the branching ratio to Z boson pairs. 95 % confidence level upper limits range from 0.53 pb at mH = 195 GeV to 0.008 pb at mH = 950 GeV for the gluon-fusion production mode and from 0.31 pb at mH = 195 GeV to 0.009 pb at mH = 950 GeV for the vector-boson-fusion production mode. The results are also interpreted in the context of Type- I and Type-II two-Higgs-doublet models.