  The Demia\'nski-Janis-Newman algorithm is an original solution generating
technique. For a long time it has been limited to producing rotating solutions,
restricting to the case of a metric and real scalar fields, despite the fact
that Demia\'nski extended it to include more parameters such as a NUT charge.
Recently two independent prescriptions have been given for extending the
algorithm to gauge fields and thus electrically charged configurations. In this
paper we aim to end setting up the algorithm by providing a missing but
important piece, which is how the transformation is applied to complex scalar
fields. We illustrate our proposal through several examples taken from N=2
supergravity, including the stationary BPS solutions from Behrndt et al. and
Sen's axion-dilaton rotating black hole. Moreover we discuss solutions that
include pairs of complex parameters, such as the mass and the NUT charge, or
the electric and magnetic charges, and we explain how to perform the algorithm
in this context (with the example of Kerr-Newman-Taub-NUT and dyonic
Kerr-Newman black holes). The final formulation of the DJN algorithm can
possibly handle solutions with five of the six Pleba\'nski-Demia\'nski
parameters along with any type of bosonic fields with spin less than two
(exemplified with the SWIP solutions). This provides all the necessary tools
for applications to general matter-coupled gravity and to (gauged)
supergravity.
