In this technical note, the fast alternating minimization algorithm (FAMA) is proposed to solve model predictive control (MPC) problems with polytopic and second-order cone constraints. Two splitting strategies with efficient implementations for MPC problems are presented. We derive computational complexity certificates for both splitting strategies, by providing complexity upper-bounds on the number of iterations required to provide a certain accuracy of the dual function value and, most importantly, of the primal solution. This is of particular relevance in the context of real-time MPC in order to bound the required on-line computation time. We further address the computation of the complexity bounds, requiring the solution of a non-convex minimization problem. Finally, we demonstrate the performance of FAMA compared to other splitting methods using a quadrotor example.