  We prove that the eigenvalues of a continuum random Schr\"odinger operator
$-\Delta+ V_{\omega}$ of Anderson type, with complex decaying potential, can be
bounded (with high probability) in terms of an $L^q$ norm of the potential for
all $q\leq d+1$. This shows that in the random setting, the exponent $q$ can be
essentially doubled compared to the deterministic bounds of Frank (Bull. Lond.
Math. Soc., 2011). This improvement is based on ideas of Bourgain (Discrete
Contin. Dyn. Syst., 2002) related to almost sure scattering for lattice
Schr\"odinger operators.
