  We investigate the Cosmic Censorship Conjecture by means of the horizon
wave-function (HWF) formalism. We consider a charged massive particle whose
quantum mechanical state is represented by a spherically symmetric Gaussian
wave-function, and restrict our attention to the superxtremal case (with
charge-to-mass ratio $\alpha>1$), which is the prototype of a naked singularity
in the classical theory. We find that one can still obtain a normalisable HWF
for $\alpha^2<{2}$, and this configuration has a non-vanishing probability of
being a black hole, thus extending the classically allowed region for a charged
black hole. However, the HWF is not normalisable for $\alpha^2 > 2$, and the
uncertainty in the location of the horizon blows up at $\alpha^2=2$, signalling
that such an object is no more well-defined. This perhaps implies that a
quantum Cosmic Censorhip might be conjectured by stating that no black holes
with charge-to-mass ratio greater than a critical value (of the order of
$\sqrt{2}$) can exist.
