  The Newman-Janis algorithm is well known to provide rotating black holes
solutions to Einstein's equations from static seeds, through a complexification
of a radial and a time coordinates. However, an ambiguity remains for the
replacement of the $r^{-1}$ and $r^{-2}$ powers of the radial coordinate. We
show here that the two cases are unified by a simple expression which allows
its extension to the $r^{2}$ power, characteristic of the de Sitter ($dS$) and
anti de Sitter ($AdS$) spacetimes. The formula leads almost automatically to
the Kerr and Kerr-Newman-$dS$ and -$AdS$ metrics.
