Due to absorption, scattering, and turbulence of water, underwater imaging often suffers from blur and low contrast, which poses a significant limitation for underwater robotic visual perception systems. In this paper, we propose a multi-task learning scheme for simultaneous image restoration and monocular depth estimation for underwater imaging, which produces a high-quality color image and a depth map with a single degraded color image captured through turbid water. The main network consists of a shared encoder and two parallel decoders, one for image restoration and the other for monocular depth estimation, enabling feature sharing across the two tasks. In addition, depth maps estimated from the degraded and restored images are fused to improve the accuracy of monocular depth estimation. The proposed method has been evaluated on a dataset of six different levels of controlled turbidity, and it outperforms four state-of-the-art deep learning models for underwater image restoration and four models for monocular depth estimation in a number of image quality metrics.