If they're not angsting about it, immortal characters have a lot they could do with the potentially infinite time on their hands. Maybe try to live a normal life, maybe shape the world as a whole, they can really get around. And some get around in a much more carnal sense. Apparently being unable to die ups your libido, as these immortals tend to have a lot more partners in one lifespan than most, with countless other lifespans to spare. Because the MayflyDecember Romance will almost certainly end in tragedy, the immortal wants to have quick flings to avoid heartbreak. Perhaps they already got around, it's just that immortality gives them much more advantage.
If the Immortal Procreation Clause in effect, that may encourage them to have as much sex without worrying about children(asides from when STD Immunity is subverted), or they may just be irresponsible and potentially father/mother hundreds of kids over hundreds-thousands of years..undoubtedly leading to a Tangled Family Tree. While Vampires Are Sex Gods is about the vampire's sex appeal, the immortality that comes with it might be why they seek out people.
If an immortal is focused on a romantic or platonic relationship, it's a case of MayflyDecember Romance and MayflyDecember Friendship. Compare Immortality Bisexuality, where eternal life encourages the immortal to swing both ways.
- Jojos Bizarre Adventure: Dio Brando becomes a vampire in Phantom Blood, and later emerges in Stardust Crusaders not having aged a day in the passing century with Jonathan Joestar's body. He is shown to have slept with countless women, later parts revealing that he's fathered four kids by different mothers, at least.
- DC Comics
- Batman: Ra's Al-Ghul is literally Really 700 Years Old, thanks to the Lazarus Pits. In that time he's had lots of kids from lots of different women, though regards none of them worthy of being his heir.
- Vandal Savage is a Contemporary Caveman 50,000 years old, and has a lot of descendants in part due to sleeping around a lot. One of his claimed identities is Genghis Khan, who's thought to have fathered at least a thousand children. This is beneficial for Vandal since his immortality is boosted and/or sustained by harvesting his descendant's organs, so he's in no worry of losing his immortality.
- The League of Extraordinary Gentlemen: Orlando is a gender-flipping 3000 year old who's been around and played a part in history. He's also The Hedonist, who's hit it off with Anything That Moves and knocked boots with a number of famous figures.
- Marvel Comics:
- The Eternals are a superhuman race that possess virtual immortality. Of them, Eros aka Starfox is a carefree and fun-loving womanizer who's Living Aphrodisiac, contrasting his nihilstic brother Thanos. Thanos used to get around in his youth, though to find some sort of satisfaction in love rather than simply trying to get his rocks off.
- X-Men: Wolverine may not literally be immortal, but with a potent Healing Factor and looking like he's 30 despite being a super-centenarian he might as well be. He's also hit it off with dozens of women over the decades despite being a grumpy, short and hair Canadian. One of his villains manages to gather a bunch of illegitimate children of his to emotionally break him by tricking Logan to off his offspring.
- Guardians of the Galaxy Vol. 2: Ego is a millions-year old Celestial that's able to manifest a Remote Body to interact with other worlds. He invokes his first purpose to mate with whatever race he comes across and sired thousands of children, with Star-Lord's mother being the only one he ever loved. This isn't just to get his rocks off; it's revealed that he needed to sire someone else with Celestial genes to complete his Assimilation Plot of the rest of the universe. Peter was the first one to meet his standards, and the rest he killed and buried enough bones to fill a cave.
- Parodied in the CollegeHumor "My Elf Girlfriend" videos: a human in an Interspecies Romance has to come to terms with the fact that his girlfriend has 2000 years more sexual history than him, including with her entire social circle in the first millennium.Ryan: Orcs?! Elowyn, orcs?!Elowyn: Everyone has a bad-boy phase in their 1200s!
- Futurama: Yivo is an immortal Eldritch Abomination who was already a trillion years old when the universe began, eventually being able to contact it after a Reality-Breaking Paradox tears a wound in reality connecting to shkler universe. The first thing shklee does is use shkler tentacles to mate with every sentient organic in the universe(except Leela) and for a time has a polygamist relationship with the 20 quadrillion inhabitants.