We extend the paradigmatic model of "complex" landscapes to complex variables.
We believe it is the first such study, a subject of interest with applications
from deep networks to optimization. In particular, we introduce an apparently
new matrix model that generalizes the well-known semicircle law for
fluctuations around real saddles in disordered systems. Our work is in line
with Bogomolny, Bohigas & Leboeuf (PRL 1992) concerning roots of one random
polynomial of high degree, while ours many of low degree. Many applications of
this problem will surely appear, as always occurs when extending a real problem
into the complex plane.