Searches are conducted for new spin-0 or spin-1 bosons using eventswhere a Higgs boson with mass 125 GeV decays into four leptons (l = e,mu). This decay is presumed to occur via an intermediate state whichcontains two on-shell, promptly decaying bosons: H -> XX/ZX 4l, wherethe new boson X has a mass between 1 and 60 GeV. The search uses ppcollision data collected with the ATLAS detector at the LHC with anintegrated luminosity of 139 fb(-1) at a centre-of-mass energy root s =13 TeV. The data are found to be consistent with Standard Modelexpectations. Limits are set on fiducial cross sections and on thebranching ratio of the Higgs boson to decay into XX/ZX, improving thosefrom previous publications by a factor between two and four. Limits arealso set on mixing parameters relevant in extensions of the StandardModel containing a dark sector where X is interpreted to be a darkboson.