The conductive bridge random access memory (CBRAM) device has been widely studied as a promising candidate for next-generation nonvolatile memory applications, where Cu as an electrode plays an important role in the resistive switching (RS) process. However, most studies only use Cu as one electrode, either the top electrode (TE) or the bottom electrode (BE); it is rarely reported that Cu is used as both TE and BE at the same time. In this study, we fabricated CBRAM devices by using Cu as both the TE and BE, and studied the RS characteristic of these devices. With Al2O3 as the switching layer (5~15 nm), the devices showed good bipolar RS characteristics. The endurance of the device could be as high as 106 cycles and the retention time could be as long as 104 s. The Al2O3 thickness influences the bipolar RS characteristic of the devices including the initial resistance, the forming process, endurance, and retention performance. The Cu electrode-based RRAM devices also present negative bias-suppressed complementary resistive switching (CRS) characteristics, which makes it effective to prevent the sneak path current or crosstalk problem in high-density memory array circuits.