  We study the critical collapse of a massive complex scalar field coupled
minimally to gravity. Taking as initial data a simple gaussian pulse with a
shape similar to the harmonic ansatz for boson stars, we obtain critical
collapse of type type I and II when varying the gaussian width $\sigma$. For
$\sigma \leq 0.5$ we find collapse of type II with a critical exponent
$\gamma=0.38\pm0.01$ and an echoing period $\Delta=3.4\pm0.1$. These values are
very similar to the known results for a real massless scalar field. On the
other hand, for $\sigma \geq 2.5$ we obtain collapse of type I. In this case we
find that the critical solutions turn out to be an unstable boson stars in the
ground state: all the data obtained from our simulations can be contrasted with
the characteristic values for unstable boson stars and their corresponding
Lyapunov exponents.
