  Critical phenomena in gravitational collapse are characterized by the
emergence of surprising structure in solution space, namely the appearance of
universal power-laws and periodicities near the threshold of collapse, and a
universal discretely self-similar solution at the threshold itself. This
seminal work spurred a comprehensive investigation of extreme spherical
spacetimes in numerical relativity, with analogous results for numerous matter
models. Recent research suggests that the generalization to less symmetric
scenarios is subtle. In twist-free axisymmetric vacuum collapse for instance,
numerical evidence suggests a breakdown of universality of solutions at the
threshold of collapse. In this study, we explore gravitational collapse
involving a massless complex scalar field minimally coupled to general
relativity. We employ the pseudospectral code BAMPS to investigate a
neighborhood of the spherically symmetric critical solution in phase space,
focusing on aspherical departures from it. First, working in explicit spherical
symmetry, we find strong evidence that the spacetime metric of the spherical
critical solution of the complex scalar field agrees with that of the Choptuik
solution. We then examine universality of the behavior of solutions near the
threshold of collapse as the departure from spherical symmetry increases,
comparing with recent investigations of the real scalar field. We present a
series of well-tuned numerical results and document shifts of the power-law
exponent and periods as a function of the degree of asphericity of the initial
data. At sufficiently high asphericities we find that the center of collapse
bifurcates, on the symmetry axis, but away from the origin. Finally we look for
and evaluate evidence that in the highly aspherical setting the collapse is
driven by gravitational waves.
