There are many significant similarities between Aemond Targaryen and Daemon in House of the Dragon, which will make their final outcomes even more satisfying.
Warning: Contains SPOILERS for House of the Dragon season 1, and George R.R. Martin’s book Fire & Blood.
Aemond Targaryen and Daemon Targaryen in House of the Dragon have several obvious connections that will have an astonishing payoff in their separate ends. In House of the Dragon, Daemon is the epitome of a rogue. He is one of the greatest warriors in the realm and he rides one of its most fearsome dragons. Daemon has long coveted the Iron Throne and he combines danger, cruelty, charm, and those flowing Valyrian locks to make him both feared and desired. Aemond would be the pretender to Daemon’s throne if it existed; much of what applies to one also applies to the other.
Particularly in season 1, episode 8, House of the Dragon overtly pushed towards the Aemond and Daemon comparisons. That caused Aemond and Daemon to temporarily argue because they both had identical hairstyles, outfits, and attitudes. It presents the two as obvious contrasts to one another; Aemond represents the greens while Daemon represents the blacks. As time goes on, that will get even better, not just as they lead the way in the Dance of the Dragons, but also for when the rivalry really heats up. Both of them will die in a fierce battle at the conclusion of it.
Aemond vs. Daemon might be the most important conflict in House of the Dragon

The parallels between Aemond and Daemon will persist as both will be essential in leading their respective sides as skilled warriors, tactical leaders, and riders of two of Westeros’ most potent dragons. But halfway during the Dance of the Dragons, during the Battle Above The God’s Eye, they will engage in combat. Daemon sends Aemond a challenge from Harrenhal while he is tearing through the Riverlands and waits almost two weeks for him to accept it. But when it is, Daemon and Caraxes clash with Aemond and Vhagar in the biggest and fiercest dragon battle of the entire civil war.
The sky is lit ablaze as the two aged, battle-weary dragons spew fire and rip at one another. This is the type of show that House of the Dragon is meant for—brutal and balletic. When done correctly, no battle should be able to compare to it in terms of thrills, surprise, and awe as Caraxes and Vahagar demonstrate their incredible strength and Aemond and Daemon their riding prowess. The story comes to an end in the most dramatic way imaginable: as the two fatally wounded dragons fall into the lake below, Daemon leaps from Caraxes and stabs himself and the young pretender with his Valyrian steel sword, Dark Sister, through the socket where his eye once was. None of them survive as they all fall into the water.
In House of the Dragon, when will Aemon vs. Daemon take place?

One of the most significant events in House of the Dragon will include Aemond and Daemon, but it will probably be some time off. The Battle Above the God’s Eye taking place at the conclusion of season 3 would make sense if George R.R. Martin’s dream for House of the Dragon to have four seasons is granted. Before then, there is a lot to get through, including significant battles and unexpected turns that may take up most of season 2 and into season 3. Thus, wrapping up Season 3 of House of the Dragon fits with its placement in the Dance of the Dragons timeline and would undoubtedly make for an epic set piece in Episode 9 or 10.