In our paper we study the extension to complex variables of the
paradigmatic
model of "complex landscape". We believe it is the first paper to study
such complex
"rugged landscapes", a subject of very high interest whose applications
range from deep networks
to optimization. In particular, we introduce and study a matrix model that
has not, to the best of our knowledge
studied previously, and which plays for the fluctuations around complex
saddles
in disordered systems the role played by the well-known semicircle law in
the real ones.
Our work is in line with the beautiful
E *Bogomolny*, O Bohigas, P *Leboeuf* - Physical Review Letters,  1992
which concerns the roots  one random polynomial of high degree, while ours
many of low degree.
We are very sure that sooner or later many new applications of this problem
will appear, as always
has been the case of extending into the complex plane the vision of a  real
problem.