Great progress has been achieved in the research field of topological states of matter during the past decade. Recently, a quasi-one-dimensional bismuth bromide, Bi₄Br₄, has been predicted to be a novel rotational-symmetry-protected topological crystalline insulator; it would also exhibit more exotic topological properties under pressure. Here, we report a thorough study of superconductivity and phase transitions in a quasi-hydrostatically pressurized a-Bi₄Br₄ crystal by combining first-principles calculations with detailed measurements of electrical resistance, alternating current magnetic susceptibility and in situ high-pressure single-crystal X-ray diffraction. We find a pressure-induced insulator-metal transition between 3.0~3.8 GPa, where valence and conduction bands cross the Fermi level to form a set of small pockets of holes and electrons. With further increase of pressure, two superconductive transitions emerge. One shows a sharp resistance drop to zero near 6.8 K at 3.8 GPa; the transition temperature gradually lowers with increasing pressure and completely vanishes above 12.0 GPa; the other sets in around 9.0 K at 5.5 GPa and persists up to the highest pressure of 45.0 GPa studied in this work. Intriguingly, we find the original superconductive phase coexists with a nontrivial rotational-symmetry-protected topology in the pressure range of 3.8~4.3 GPa; the second one is associated with a structural phase transition from monoclinic C2/m to triclinic P-1 symmetry.