Based on the definitions of fractional-order differential and Adomian decomposition algorithm, the numerical solution of the fractional-order simplified Lorenz system is investigated. Results show that compared with the Adams-Bashforth-Moulton algorithm, Adomian decomposition algorithm yields more accurate results and needs less computing as well as memory resources. It is even more accurate than Runge-Kutta algorithm when solving the integer order system. The minimum order of the simplified Lorenz system solved by using Adomian decomposition algorithm is 1.35, which is much smaller than 2.79 achieved by the Adams-Bashforth-Moulton algorithm. Dynamical characteristics of the system are studied by the phase diagram, bifurcation analysis, and complexities are calculated by employing the spectral entropy (SE) algorithm and C0 algorithm. Complexity results are consistent with the bifurcation diagrams, for which mean complexity can also reflect the dynamic characteristics of a chaotic system. Complexity decreases with increasing order q, and there are little influences on complexity versus changes of parameter c when the system is chaotic. It provides a theoretical and experimental basis for the application of fractional-order chaotic system in the field of encryption and secure communication.