In 2012, a particle with mass of 125 GeV was observed by the Atlas and CMS collaborations at the Large Hadron Collider, consistent with the Higgs boson predicted by the Brout-Englert-Higgs mechanism of electroweak-symmetry breaking. However, various fundamental questions in the standard model of particle physics (SM) remain unanswered. With the existence of the Higgs boson confirmed, the SM now faces a theoretical challenge: large known quantum corrections to the Higgs mass turn out to nearly perfectly cancel each other in practice, without explanation. Various proposed extensions of the SM that introduce new fundamental physics at the TeV energy scale provide such an explanation and stabilize the theory up to the energy scale of the Big Bang. One of the proposed solutions results in five types of Higgs bosons, which may strongly couple to b-quarks. I propose to search for the H->ZA decay, a signature that is still largely unexplored. Secondly, evidence for the existence of dark matter from cosmological observations is abundant, yet the SM provides no particle that has the properties of dark matter. I propose to search for its production in scalar-boson decays; would dark matter couple only to the scalar sector, decays of the Higgs boson could probe for the first time directly its interactions. Thirdly, extensions of the SM do not only predict the existence of new particles, but will also subtly modify the properties of the observed 125 GeV particle. New precision studies of the decay of this boson into quarks will provide a complementary strategy to search for new physics. I propose to address several of these questions by exploring a single LHC collision topology: the rare production of Z bosons in association with b jets. Using my expertise in this topology, while simultaneously targeting three research topics, I will address fundamental questions.