We use new near-infrared spectroscopic observations to investigate the nature and evolution of the most luminous H\alpha (Ha) emitters at z~0.8-2.23, which evolve strongly in number density over this period, and compare them to more typical Ha emitters. We study 59 luminous Ha emitters with $L_{H\alpha}>L_{H\alpha}^*$, roughly equally split per redshift slice at z~0.8, 1.47 and 2.23 from the HiZELS and CF-HiZELS surveys. We find that, overall, 30$\pm$8% are AGN (80$\pm$30% of these AGN are broad-line AGN, BL-AGN), and we find little to no evolution in the AGN fraction with redshift, within the errors. However, the AGN fraction increases strongly with Ha luminosity and correlates best with $L_{H\alpha}/L_{H\alpha}^*(z)$. While $L_{H\alpha} 80%), the most luminous Ha emitters ($L_{H\alpha}>10L_{H\alpha}^*(z)$) at any cosmic time are essentially all BL-AGN. Using our AGN-decontaminated sample of luminous star-forming galaxies, and integrating down to a fixed Ha luminosity, we find a factor of ~1300x evolution in the star formation rate density from z=0 to z=2.23. This is much stronger than the evolution from typical Ha star-forming galaxies and in line with the evolution seen for constant luminosity cuts used to select "Ultra-Luminous" Infrared Galaxies and/or sub-millimetre galaxies. By taking into account the evolution in the typical Ha luminosity, we show that the most strongly star-forming Ha-selected galaxies at any epoch ($L_{H\alpha}>L^*_{H\alpha}(z)$) contribute the same fractional amount of ~15% to the total star-formation rate density, at least up to z=2.23.