The complex dynamics analysis of fractional-order neural networks is a cutting-edge topic in the field of neural network research. In this paper, a fractional-order Hopfield neural network (FOHNN) system is proposed, which contains four neurons. Using the Adomian decomposition method, the FOHNN system is solved. The dissipative characteristics of the system are discussed, as well as the equilibrium point is resolved. The characteristics of the dynamics through the phase diagram, the bifurcation diagram, the Lyapunov exponential spectrum, and the Lyapunov dimension of the system are investigated. The circuit of the system was also designed, based on the Multisim simulation platform, and the simulation of the circuit was realized. The simulation results show that the proposed FOHNN system exhibits many interesting phenomena, which provides more basis for the study of complex brain working patterns, and more references for the design, as well as the hardware implementation of the realized fractional-order neural network circuit.