  We study perturbations of a complex scalar field during reheating with no
self-interaction in the regime $ \mu \gg H$, when the scalar field has a fast
oscillatory behaviour (close to a pressure-less fluid). We focus on the precise
determination of the instability scale and find it differs from that associated
with a real scalar field. We further look at the probability that unstable
fluctuations form Primordial Black Holes (PBHs) obtaining a significant
production of tiny PBHs which quickly evaporate and may subsequently leave a
population of Planck-mass relics. We finally impose restrictions on the
duration and energy scale of the fast oscillations period by considering that
such relics constitute, at most, the totality of dark matter in the Universe.
