The random quantum Ashkin-Teller chain is studied numerically by means of
time-dependent Density-Matrix Renormalization Group. The critical lines are
estimated as the location of the peaks of the integrated autocorrelation times,
computed from spin-spin and polarization-polarization autocorrelation
functions. Disorder fluctuations of magnetization and polarization are observed
to be maximum on these critical lines. Entanglement entropy leads to the same
phase diagram, though with larger Finite-Size effects. The decay of spin-spin
and polarization-polarization autocorrelation functions provides numerical
evidence of the existence of a double Griffiths phase when taking into account
finite-size effects. The two associated dynamical exponents z increase rapidly
as the critical lines are approached, in agreement with the recent conjecture
of a divergence at the two transitions in the thermodynamic limit
