While traveling between dimensions during one of his more bizarre adventures, Nightcrawler and his traveling companion the Vanisher glimpsed several alternate versions of themselves. Kurt caught glimpses of himself as a female, a duck and a robot.
Trying to rescue Illyana Rasputin from her kidnapper, Belasco, an alternate team of X-Men became stranded in the sorcerer's hellish Limbo. While some of their teammates were killed, Nightcrawler ended up becoming a loyal servant of Belasco, whereas Kitty Pryde was turned into a cat creature, Cat, but remained free. Later on, while fighting over Illyana, Cat phased Nightcrawler's foot into the ground, effectively trapping him so that she could kill him with her sword. Much later, a similar Nightcrawler was again seen in Belasco's servitude, though it's unknown whether the sorcerer resurrected his lackey or if he merely made use of another counterpart of Nightcrawler.
At one point, Loki offered the X-Men and the New Mutants to stay in Asgard, though he forced all to make the same decision and the ones willing to stay felt that they couldn’t demand from their teammates to abandon Earth. In the alternate reality depicted in What if? (2nd series) #12, Loki allowed them each to chose individually and Nightcrawler was among those to accept the invitation to stay. Asgard was a dream come true for Kurt, as he could now be the daring swashbuckler that he had always wanted to be. Eventually, he even joined the fabled Warriors Three, him being the replacement for Fandrall, who had fallen in love with Rogue and wanted to retire after marrying her.
In the alternate timeline shown in What if (2nd series) #23, the original X-Men managed to defeat Krakoa all by themselves, and Xavier had no need to recruit Nightcrawler and the other international X-Men. Instead, Kurt was saved from the angry mob chasing him by someone else – Shi'ar agent Davan Shakari, in the guise of Eric the Red. 
Next to all humans being afraid of mutants and having treated him rather hostile, Nightcrawler's hatred towards them was easily fueled. He joined the alien's fight against the X-Men and, following their defeat, he was the only one to avoid capture. Plotting revenge, Nightcrawler secretly followed the X-Men's every move for weeks, during which he slowly understood their just cause. Nightcrawler finally made his move when the X-Men were alerted by the Shi'ar to an ancient force, the so-called Chaos-Bringer, residing in Earth's sun. The only way to destroy it was for it to merge with a willing host, who could then be killed. Cyclops wanted to make the noble sacrifice, but Nightcrawler knocked him out and took his place on the shuttle. By doing so, Kurt hoped to atone for his past mistakes. Additionally, he experienced something incredibly beautiful in the few seconds between merging with the force and being disintegrated by the Shi'ar's powerful weapons.
In the multiverse, there are several timelines where Nazi Germany won World War II. One of them, Earth-597, had a version of Excalibur too, though they were called Lightning Force. The line-up included Meggan, Shadowcat, Hauptmann Englande (Brian Braddock) and Nightcrawler. This version of Kurt was a bit cocky and over-confident, and apparently a smoker. Whereas Kurt’s usually rather charming with the lovely ladies, this twisted counterpart of him wouldn’t take No for an answer and tried to rape at least one woman.
There were numerous other versions of Nightcrawler (and Excalibur) that were introduced either during the Cross-Time Caper or popped up near the main universe Excalibur's lighthouse, as it was a nexus place. Most of them were unrecognizable and seemed to be based on sentient animal life forms. Kurt's counterpart of a reality inhabited by sentient dinosaurs called himself "Nightstrutter." Among the few humanoid counterparts, there was one version of Kurt that showed no signs of being a mutant at all, him looking entirely human. Him being from an Earth where super-powered people were hunted down and genetically changed into baseline humans, it's rather likely that he underwent such a process himself. However, it didn't stop him from becoming one of the Justicers, the police officers enforcing this law.
Nightcrawler married his childhood sweetheart Amanda Sefton in the alternate future timeline known as Days of Future Past. They had at least one child that is known of, 
the teleporter named Blue, who inherited her father's unique appearance and operated as a member of the New Mutants under Banshee's tutelage. Not much else is known about Kurt's life in the future, besides the fact the he was among the first to die when the government outlawed the X-Men. Nightcrawler and Amanda were helping Illyana to her school bus from the mansion when Project: Wideawake launched an assault on the Xavier estate. Those three were killed in the initial salvo.
In the Age of Apocalypse, Nightcrawler was raised as Kurt Darkholme, and reconnected with his mother much earlier in his life than his 616 counterpart. This Kurt disdained all religions and was far more brutal with his teleportation powers, using them dismember and decapitate his victims. After learning that his timeline might be "wrong," Nightcrawler worked with the X-Men to restore reality to its proper form, even at the expense of his own existence. However, his timeline persevered, and he met X-Force from Earth-616 much later when they traveled to the AoA to save their own world. After completing their mission, Kurt decided to stay in the 616 dimension and join X-Force so he could track down enemies from his native world who had hopped timelines. On Earth-616, he began eliminating the targets on his list, but to get to his main target, he had to betray his X-Force teammates. Although he succeeded in getting revenge, X-Force did not take the betrayal kindly, and Kurt had to flee before he could explain. He fled and returned home, only to discover his entire dimension was under attack by ancient cosmic entities. Kurt made amends with the people he had hurt before sacrificing his life to seal off the Age of Apocalypse from the rest of existence in order to contain the existential threat of its invaders.
For more information, see UXN's Spotlight On... Nightcrawler (AoA).
Mystique also had chosen to not give up her son in the alternate reality shown in What If (2nd series) #98. Naming him Michael, Mystique kept her son secret from the public, even from her teammates in the Brotherhood of Evil Mutants, fearing that he would be hunted down for his demonic appearance.
Later, Mystique adopted Rogue, who snooped around the house and eventually found Michael hiding in the attic of the house. They became friends and planned to flee but they were confronted by the Brotherhood, who, unaware of Nightcrawler's existence, assumed he was an X-Man being sent to attack them. Avalanche, Blob and Pyro ended up paralyzing Michael in battle and Michael told Rogue to go on for them, to be free, because he was too weak to teleport them to safety. Rogue wouldn't hear any of that, though, and deeply kissed him, absorbing the last of his powers in order to escape. However, the absorption became permanent, and not only did Rogue keep the demonic appearance but also the personality of her adoptive brother.