  In the case of vanishing cosmological constant, Demia\'nski has shown that
the Janis-Newman algorithm can be generalized in order to include a NUT charge
and another parameter $c$, in addition to the angular momentum. Moreover it was
proved that only a NUT charge can be added for non-vanishing cosmological
constant. However despite the fact that the form of the coordinate
transformations was obtained, it was not explained how to perform the
complexification on the metric function, and the procedure does not follow
directly from the usual Janis-Newman rules. The goal of our paper is threefold:
explain the hidden assumptions of Demia\'nski's analysis, generalize the
computations to topological horizons (spherical and hyperbolic) and to charged
solutions, and explain how to perform the complexification of the function. In
particular we present a new solution which is an extension of the Demia\'nski
metric to hyperbolic horizons. These different results open the door to
applications in (gauged) supergravity since they allow for a systematic
application of the Demia\'nski-Janis-Newman algorithm.
