We derive an exact radiating Kerr-Newman like black hole solution, with constant curvature R= R 0 imposed, to metric f( R) gravity via complex transformations suggested by Newman-Janis. This generates a geometry which is precisely that of radiating Kerr-Newman-de Sitter/anti-de Sitter with the f( R) gravity term R 0 contributing a cosmological-like term. The structure of three horizon-like surfaces, viz. time-like limit surface, apparent horizon, and event horizon, are determined. We demonstrate the existence of an additional cosmological horizon, in f( R) gravity model, apart from the regular black hole horizons that exist in the analogous general relativity case. In particular, the known stationary Kerr-Newman black hole solutions of f( R) gravity and general relativity are retrieved. We find that the time-like limit surface becomes less prolate with R 0 thereby affecting the shape of the corresponding ergosphere.