The Avatar of Khaine is the final Eldar boss. This mission is available during the turn you can take the mission The Last Stand. The Avatar is also extremely hared to defeat, and has around 290,000 HP. Use your strategic asset to the left, or the reinforcement beacon to the right as a retreating point. The Avatar is very susceptible to plasma fire, so try keeping all your squads on the ledge before the stairs and keep Avitus' fire at the Avatar. You do not even have to destroy the Avatar's support units. Alternate your squads between attacking and retreating.
For both final bosses, try using Orbital Bombardments, artillery strikes, and other 'end-game' special accessories. These can take a chunk of health out of the extremely resistant bosses. Both bosses can take around 20 minutes of fighting each to kill.