We report a joint experimental and theoretical study of the structural, vibrational, and electric properties of the topological insulator PbBi2Te4 under compression through X-ray diffraction, Raman scattering, and electrical measurements at high pressure, which are complemented with theoretical calculations that include the analysis of the topological electron density. The internal polyhedral compressibility of the rhombohedral phase, the behavior of its Raman-active modes, the electrical behavior, and the nature of its different bonds under compression are discussed and compared with their parent binary compounds and with related ternary materials. Interestingly, PbBi2Te4 retains its topological insulating properties as far as the rhombohedral tetradymite-like phase is retained under compression, unlike other tetradymite-like compounds. In addition, PbBi2Te4 undergoes an unconventional reversible pressure-induced decomposition into the high-pressure phases of its parent binary compounds (PbTe and Bi2Te3) above 8 GPa. Finally, we discuss that the intralayer bonds in the tetradymite-like structure of PbBi2Te4 are electron-deficient multicenter bonds; i.e. bonds of the same kind as those present in its parent binary compounds, in related chalcogenides, such as SnSb2Te4, and in general in chalcogenide-based phase change materials. These unconventional bonds are intermediate between covalent and metallic bonds and provide exceptional properties to the materials.