Compound dry hot events (CDHEs), where hot events and droughts coexist, have received a lot of attention lately due to their catastrophic effects on the economy, environment and human health. In this study, we use two CDHE indices, the Standardized Compound Event Indicator (SCEI) and the Standardized Dry and Hot Index (SDHI), to assess changes in CDHE characteristics (severity, frequency, spatial extent) over the historical past and future CMIP6 simulations across the Indian subcontinent. To understand the role of the drought index selected on CDHE characterization two drought indices namely the Standardized Precipitation Index (SPI) and the Standardized Precipitation Evapotranspiration Index (SPEI) are employed in the calculation of the CDHE indices. Further, the role of climatic oscillations such as El Nino Southern Oscillation (ENSO), Pacific Decadal Oscillation (PDO), Equatorial Indian Ocean Monsoon Oscillation and Indian Ocean Dipole in modulating CDHE characteristics have also been explored. Results show that SPI and SPEI based CDHE indices exhibit contrasting trends in northern India, while similar trends are observed in southern India in the historical past. Also, more frequent, extended, and severe CDHEs are reported by CDHE indices calculated using SPEI than by SPI. Temperature is found to be the dominant factor contributing to increases in CDHEs in the recent past and ENSO phases significantly modulate the severity and frequency of CDHE events in India. CMIP6 simulations generally report an increase in CDHE events for a 3°C global warming scenario. Overall, our findings show that the choice of the drought index has a greater impact on CDHE characterization than the choice of the CDHE index itself. Results from this study provide useful information towards understanding the risk of CDHEs in India under global warming and urge for the development and implementation of adaptation and mitigation measures.