  In this paper we are concerned with determining lower bounds of the number of
limit cycles for piecewise polynomial holomorphic systems with a straight line
of discontinuity. We approach this problem with different points of view: study
of the number of zeros of the first and second order averaging functions, or
with the control of the limit cycles appearing from a monodromic equilibrium
point via a degenerated Andronov-Hoph type bifurcation, adding at the very end
the sliding effects. We also use the Poincar\'e-Miranda theorem for obtaining
an explicit piecewise linear holomorphic systems with 3 limit cycles, result
that improves the known examples in the literature that had a single limit
cycle.
