  We propose new superintegrable mechanical system on the complex projective
space $\mathbb{CP}^N$ involving a potential term together with coupling to a
constant magnetic fields. This system can be viewed as a $\mathbb{CP}^N$-analog
of both the flat singular oscillator and its spherical analog known as
"Rosochatius system". We find its constants of motion and calculate their
(highly nonlinear) algebra. We also present its classical and quantum
solutions. The system belongs to the class of "K\"ahler oscillators" admitting
$SU(2|1)$ supersymmetric extension. We show that, in the absence of magnetic
field and with the special choice of the characteristic parameters, one can
construct $\mathcal{N}=4, d=1$ Poinacar\'e supersymmetric extension of the
system considered.
