Roadside camera-driven 3D object detection is a crucial task in intelligent transportation systems, which extends the perception range beyond the limitations of vision-centric vehicles and enhances road safety. While previous studies have limitations in using only depth or height information, we find both depth and height matter and they are in fact complementary. The depth feature encompasses precise geometric cues, whereas the height feature is primarily focused on distinguishing between various categories of height intervals, essentially providing semantic context. This insight motivates the development of Complementary-BEV (CoBEV), a novel end-to-end monocular 3D object detection framework that integrates depth and height to construct robust BEV representations. In essence, CoBEV estimates each pixel's depth and height distribution and lifts the camera features into 3D space for lateral fusion using the newly proposed two-stage complementary feature selection (CFS) module. A BEV feature distillation framework is also seamlessly integrated to further enhance the detection accuracy from the prior knowledge of the fusion-modal CoBEV teacher. We conduct extensive experiments on the public 3D detection benchmarks of roadside camera-based DAIR-V2X-I and Rope3D, as well as the private Supremind-Road dataset, demonstrating that CoBEV not only achieves the accuracy of the new state-of-the-art, but also significantly advances the robustness of previous methods in challenging long-distance scenarios and noisy camera disturbance, and enhances generalization by a large margin in heterologous settings with drastic changes in scene and camera parameters. For the first time, the vehicle AP score of a camera model reaches 80% on DAIR-V2X-I in terms of easy mode. The source code will be made publicly available at https://github.com/MasterHow/CoBEV.