To address the rapidly increasing challenges posed by the demands for large-scale data transmission and processing, this paper proposes a high-data-rate hybrid index modulation communication system based on quadrature chaos shift keying (HR-HIM-QCSK). The system employs a hybrid modulation involving carrier index and slot index. In this approach, three mutually orthogonal reference signals are constructed using Walsh codes. These reference signals are then utilized to create three distinct QCSK signals, distinguishing between the activation modes of carriers and time slots. This design allows all slots to carry QCSK signals, significantly enhancing the data rate of the HR-HIM-QCSK system. Theoretical analysis and simulations conducted under additive white Gaussian noise and multipath Rayleigh fading channels demonstrate that the HR-HIM-QCSK system outperforms other competing systems in terms of bit error rate (BER) performance. Further analysis covers aspects such as data rate, spectral efficiency, complexity and covertness, highlighting the superior performance of the HR-HIM-QCSK system in high-data-rate communication. This study provides valuable insights for the design of efficient chaotic communication systems.