We propose a novel two-stage training strategy with ambiguity boosting for the self-supervised learning of single view depths from stereo images. Our proposed two-stage learning strategy first aims to obtain a coarse depth prior by training an auto-encoder network for a stereoscopic view synthesis task. This prior knowledge is then boosted and used to self-supervise the model in the second stage of training in our novel ambiguity boosting loss. Our ambiguity boosting loss is a confidence-guided type of data augmentation loss that improves the accuracy and consistency of generated depth maps under several transformations of the single-image input. To show the benefits of the proposed two-stage training strategy with boosting, our two previous depth estimation (DE) networks, one with t-shaped adaptive kernels and the other with exponential disparity volumes, are extended with our new learning strategy, referred to as DBoosterNet-t and DBoosterNet-e, respectively. Our self-supervised DBoosterNets are competitive, and in some cases even better, compared to the most recent supervised SOTA methods, and are remarkably superior to the previous self-supervised methods for monocular DE on the challenging KITTI dataset. We present intensive experimental results, showing the efficacy of our method for the self-supervised monocular DE task.