With the rapid development of network science, Turing patterns on complex networks have attracted extensive attention from researchers. In this paper, we focus on spatial patterns in multiplex ER (Erdös-Rényi) random networks, taking the predator–prey model with Allee effect and hyperbolic mortality as an example. In theory, the threshold condition for generating Turing patterns is given using the Turing instability theory of multiplex networks. Numerically, we design relevant experiments to explore the impact of network topology on Turing patterns. The factors considered include model parameters, diffusion rate, average degree of the network, and differences in the average degree of different layers. The results indicate that the importance of diffusion rate and network average degree for Turing patterns is affirmed on the single-layer network. For multiplex networks, the differentiation of average degrees in different layers controls the generation of Turing patterns, which are not affected by the diffusion rates of the two populations. More interestingly, we observe the switching of Turing patterns and spatiotemporal patterns. We believe that these findings contribute to a better understanding of self-organization on complex networks.