A theory describing above-threshold ionization of atoms and ions in a strong electromagnetic field is presented. It is based on the widely known strong field approximation and incorporates the Coulomb interaction between the photoelectron and the nucleus using the method of complex classical trajectories. A central result of the theory is the Coulomb-corrected ionization amplitude whose evaluation requires little extra numerical effort. By comparing our predictions with the results of ab initio numerical solutions for two examples we show that the new theory provides a significant improvement of the Coulomb-free strong field approximation. For the case of above-threshold ionization in elliptically polarized fields a comparison with available experimental data is also presented.