The 3D High Efficiency Video Coding (3D-HEVC) standard aims to code 3D videos that usually contain multi-view texture videos and its corresponding depth information. It inherits the same quadtree prediction structure of HEVC to code both texture videos and depth maps. Each coding unit (CU) allows recursively splitting into four equal sub-CUs. At each CU depth level, it enables 10 types of inter modes and 35 types of intra modes in inter frames. Furthermore, the inter-view prediction tools are applied to each view in the test model of 3D-HEVC (HTM), which uses variable size disparity-compensated prediction to exploit inter-view correlation within neighbor views. It also exploits redundancies between a texture video and its associated depth using inter-component coding tools. These achieve the highest coding efficiency to code 3D videos but require a very high computational complexity. In this paper, we propose a context-adaptive based fast CU processing algorithm to jointly optimize the most complex components of HTM including CU depth level decision, mode decision, motion estimation (ME) and disparity estimation (DE) processes. It is based on the hypothesis that the optimal CU depth level, prediction mode and motion vector of a CU are correlated with those from spatiotemporal, inter-view and inter-component neighboring CUs. We analyze the video content based on coding information from neighboring CUs and early predict each CU into one of five categories i.e., DE-omitted CU, ME-DE-omitted CU, SPLIT CU, Non-SPLIT CU and normal CU, and then each type of CU adaptively adopts different processing strategies. Experimental results show that the proposed algorithm saves 70% encoder runtime on average with only a 0.1% BD-rate increase on coded views and 0.8% BD-rate increase on synthesized views. Our algorithm outperforms the state-of-the-art algorithms in terms of coding time saving or with better RD performance.