Hyperbolic materials have wide application prospects, such as all-angle negative refraction, sub-diffraction imaging and nano-sensing, owning to the unusual electromagnetic response characteristics. Compared with artificial hyperbolic metamaterials, natural hyperbolic materials have many advantages. Anisotropic two-dimensional (2D) materials show great potential in the field of optoelectronics due to the intrinsic in-plane anisotropy. Here, the electronic and optical properties of two hyperbolic 2D materials, monolayer CuB6 and CuB3, are investigated using first-principles calculations. They are predicted to have multiple broadband hyperbolic windows with low loss and highly-anisotropic plasmon excitation from infrared to ultraviolet regions. Remarkably, plasmon propagation along the x-direction is almost forbidden in CuB3 monolayer. The hyperbolic windows and plasmonic properties of these 2D copper borides can be effectively regulated by electron (or hole) doping, which offers a promising strategy for tuning the optical properties of the materials.