  We derive an exact radiating Kerr-Newman like black hole solution, with
constant curvature $R=R_0$ imposed, to {\it 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 contributing an $R_0$ cosmological-like term. The structure of
three horizon-like surfaces, {\it viz.} timelike 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 timelike
limit surface becomes less prolate with $R_0$ thereby affecting the shape of
the corresponding ergosphere.
