Matrix completion-assisted channel estimation is considered one of promising techniques in millimeter wave (mmWave) massive multiple input multiple output (MIMO) system by exploiting the low-rank property of channel matrix in the angle domain. However, existing channel estimation approaches are hard to achieve high accuracy due to the inevitable bias solution caused by nuclear norm based minimization (NNM). To address this problem, this paper proposes a novel matrix completion-assisted mmWave massive MIMO channel estimation method. We employ an effective and flexible rank function named joint weighted and truncated nuclear norm as relaxation of nuclear norm, and then construct an novel matrix completion model for channel estimation problem. Moreover, a popular framework of alternating direction method of multipliers (ADMM) is derived for minimization of the resulting optimization problem. Simulation results are provided to verify the proposed method that can flexibly and effectively improve the channel estimation accuracy with reliable convergence.