Systems with hidden attractors have been the hot research topic of recent years because of their striking features. Fractional-order systems with hidden attractors are newly introduced and barely investigated. In this paper, a new 4D fractional-order chaotic system with hidden attractors is proposed. The abundant and complex hidden dynamical behaviors are studied by nonlinear theory, numerical simulation, and circuit realization. As the main mode of electrical behavior in many neuroendocrine cells, bursting oscillations (BOs) exist in this system. This complicated phenomenon is seldom found in the chaotic systems, especially in the fractional-order chaotic systems without equilibrium points. With the view of practical application, the spectral entropy (SE) algorithm is chosen to estimate the complexity of this fractional-order system for selecting more appropriate parameters. Interestingly, there is a state variable correlated with offset boosting that can adjust the amplitude of the variable conveniently. In addition, the circuit of this fractional-order chaotic system is designed and verified by analog as well as hardware circuit. All the results are very consistent with those of numerical simulation.