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While 360° images are becoming ubiquitous due to popularity of panoramic content, they cannot directly work with most of the existing depth estimation techniques developed for perspective images. In this paper, we present a deep-learning-based framework of estimating depth from 360° images. We present an adaptive depth refinement procedure that refines depth estimates using normal estimates and pixel-wise uncertainty scores. We introduce double quaternion approximation to combine the loss of the joint estimation of depth and surface normal. Furthermore, we use the double quaternion formulation to also measure stereo consistency between the horizontally displaced depth maps, leading to a new loss function for training a depth estimation CNN. Results show that the new double-quaternion-based loss and the adaptive depth refinement procedure lead to better network performance. Our proposed method can be used with monocular as well as stereo images. When evaluated on several datasets, our method surpasses state-of-the-art methods on most metrics.