In this paper, we propose a deep learning-based signal detector called DuaIM-3DNet for dual-mode index modulation-based three-dimensional (3D) orthogonal frequency division multiplexing (DM-IM-3D-OFDM). Herein, DM-IM-3D- OFDM is a subcarrier index modulation scheme which conveys data bits via both dual-mode 3D constellation symbols and indices of active subcarriers. Thus, this scheme obtains better error performance than the existing IM schemes when using the conventional maximum likelihood (ML) detector, which, however, suffers from high computational complexity, especially when the system parameters increase. In order to address this fundamental issue, we propose the usage of a deep neural network (DNN) at the receiver to jointly and reliably detect both symbols and index bits of DM-IM-3D-OFDM under Rayleigh fading channels in a data-driven manner. Simulation results demonstrate that our proposed DNN detector achieves near-optimal performance at significantly lower runtime complexity compared to the ML detector.