In a recent paper we introduced a trust-region method with variable norms for unconstrained minimization and we proved standard asymptotic convergence results. Here we will show that, with a simple modification with respect to the sufficient descent condition and replacing the trust-region approach with a suitable cubic regularization, the complexity of this method for finding approximate first-order stationary points is $O(\varepsilon^{-3/2})$. Some numerical experiments are also presented to illustrate the impact of the modification on practical performance.