Presented is the search for anomalous Higgs boson decays into two axion-like particles (ALPs) using the full Run 2 data set of 140 fb(-1) of proton-proton collisions at a centre-of-mass energy of 13 TeV recorded by the ATLAS experiment. The ALPs are assumed to decay into two photons, providing sensitivity to recently proposed models that could explain the (g -2)(mu) discrepancy. This analysis covers an ALP mass range from 100 to 62 GeV and ALP-photon couplings in the range 10(-7) TeV-1 < C-a gamma gamma/Lambda < 1 TeV-1, and therefore includes signatures with significantly displaced vertices and highly collinear photons. No significant excess of events above the Standard Model background is observed. Upper limits at 95% confidence level are placed on the branching ratio of the Higgs boson to two ALPs in the four-photon final state, and are in the range of 10(-5) to 3x10(-2), depending on the hypothesized ALP mass and ALP-photon coupling strength.