The goal of this project was to search for new beyond the Standard Model (SM) physics in the Higgs sector, through a novel search for anomalous decays of the Higgs boson into a Z boson and an undiscovered light neutral boson. The necessity for such new physics is motivated by several fundamental questions, such as the unexplained nature of the dark matter, that corresponds to 85% of the matter of the universe, and the origin of the fermion mass hierarchy, which spans several orders of magnitude in mass. The SM offers no explanations for these important questions. Several ideas put forward to explain these puzzles, such as those involving extended Higgs sectors, also predict the existence of new light neutral bosons, often denoted a. Such bosons are expected to be produced in the decays of the observed Higgs boson, either in pairs, H -> aa, or in association with a Z boson, H -> Za. While the H -> aa channel had been searched for at the LHC, until the completion of this project, the H -> Za channel represented a highly viable yet largely under-investigated probe of potential new physics in the Higgs sector requiring urgent attention. In this project a direct search was performed for the the H -> Za decay with proton-proton collision data collected by the ATLAS experiment during Run 2 of the CERN Large Hadron Collider. The experimental strategy was designed to maximise the sensitivity by targeting boosted hadronic decays of the a boson. This goal was achieved by the development of several novel tools. No excess of events over the expected background was found, and an upper limit on the fraction of these potential decays of the Higgs boson of around 30% was set at 95% confidence level. The results of this seminal study were published in Phys. Rev. Lett. 125 (2020) 22 (arXiv:2004.01678).