Exploring complex social contagion, especially the social reinforcement dynamics when exposing to multiple sources, is of vital importance for understanding the spread of complicated collective behaviors nowadays. While previous works exclusively focus on single-layer networks, the detailed impacts of multilayer reinforcement which characterizes the promoting effects among multiple social circles remain largely unknown. To this end, we incorporate multilayer reinforcement into ignorant-spreader-ignorant (SIS) model on multiplex networks. Our theoretical analysis combines pairwise method and mean-field theory and agrees well with large-scale simulations. Surprisingly, we find this complex social contagion mechanism triggers the emergence of bistability phenomena, where extinction and outbreak states coexist. Further, we show that the final state of bistable regions depends on the initial density of adopters, the critical value of which decreases as the contagion transmissibility or the multilayer reinforcement increases. In particular, we highlight two possible conditions for the outbreak of social contagion: to possess large contagion transmissibility, or to possess large initial density of adopters with strong multilayer reinforcement. Our results show the powerful and non-negligible impacts of complex dynamical mechanisms, which provides valuable insights toward the spreading behaviors in the digital age.