Self-assembling organic polymers and copper-oxide compounds are two classes of unconventional superconductors, whose challenging behavior does not comply with the traditional picture of Bardeen-Cooper-Schrieffer (BCS) superconductivity in regular crystals. In this paper, we propose a theoretical model that accounts for the basic superconducting properties of either class of the unconventional materials. These properties are considered as interconnected manifestations of the same phenomenon: We argue that superconductivity occurs in both cases because the charge carriers (i.e., electrons or holes) exchange fracton excitations, quantum oscillations of fractal lattices that mimic the complex microscopic organization of the unconventional superconductors. For the copper oxides, the superconducting transition temperature T-c as predicted by the fracton mechanism is of the order of similar to150 K. We suggest that the marginal ingredient of the high-temperature superconducting phase is provided by fracton coupled holes that condensate in the conducting copper-oxygen planes owing to the intrinsic field-effect-transistor configuration of the cuprate compounds. For the gate-induced superconducting phase in the electron-doped polymers, we simultaneously find a rather modest transition temperature of similar to2-3 K owing to the limitations imposed by the electron tunneling processes on a fractal geometry. We speculate that the hole-type superconductivity shows a larger onset temperature when compared to its electron-type counterpart. This promises an intriguing possibility of high-temperature superconducting states in hole-doped complex materials. The theoretical methods applied in our study bring together the so-called strange (or fractional) dynamics and the unconventional, topological description of the complex fractal sets underlying the fracton spectrum. A generalized kinetic equation containing integer time and fractional real-space derivatives is found for the fracton excitations in the harmonic approximation. The fracton superconductivity mechanism is further discussed in connection with experimental observations. An important prediction of the present study is the universality of ac conduction of the unconventional materials above the superconducting transition temperature T-c.