Vanya Becomes The White Violin
An interesting detail in The Umbrella Academy's climactic final scenes is that, as Vanya performs her solo and unleashes the full potential of her powers, the suit she is wearing and the violin she inherited from Sir Reginald turn white. This is a nod to the comics by Gerard Way and Gabriel Bá, upon which The Umbrella Academy is based. In the miniseries "The Apocalypse Suite" - the direct influence for The Umbrella Academy season 1 - Vanya is captured by a villainous orchestra and transformed by the Conductor using Hargreeves' notes. Her skin, eyes and hair turn deathly white and she is given a white violin, which she uses to channel her powers.
In the TV show, obviously, things are slightly different. While Harold Jenkins serves as a villain in the latter half of the season, serving a similar role to the Conductor by teaching Vanya how to unleash her powers using Reginald Hargreeves' notes, Vanya ultimately makes the final transition to full-blown villainy by herself.
Klaus' New Ability Is A Game-Changer
Like Vanya, Klaus' abilities were suppressed - not by Sir Reginald, who was eager for Klaus to explore his full potential, but by Klaus himself. Sir Reginald tried to bring Klaus' powers to the forefront and cure him of his fear of the dead by locking him up in a mausoleum all night, but all he succeeded in doing was traumatizing Klaus so that, as an adult, he went to great lengths to avoid seeing the dead. Just as Vanya's anxiety medication kept her docile and stopped her from being able to use her powers, Klaus' constant abuse of drugs and alcohol dampened his abilities until the only ghost he could consistently see was Ben.
When Klaus finally manages to get sober in an effort to see the love of his life again, the chemical haze that has been blocking his full potential is finally lifted. He doesn't have complete control of it just yet, and it seems to be triggered by a rush of panic or adrenaline. The three times that we see Klaus use his new powers are when he gets in a fight with Ben and tries to get high again, when he sees the ceiling falling and realizes he and Diego are about to die, and when he and his siblings are pinned down by The Commission's agents. During this final manifestation of his powers we see Klaus appear to exert some measure of control, his hands glowing blue as he gives Ben a physical form.
This isn't quite the same as being able to bring the dead back to life, but it's definitely a game-changer. Klaus can now not only see ghosts himself - he can also give them a physical form so that others can see them as well. More importantly, his new ability means that Ben is off the bench and able to take part in fights again, using his formidable abilities. That is... if Ben is still dead.
Where Did The Umbrella Academy Time Travel To?
One of the most intriguing aspects of the circling shot that shows the de-aged members of the Umbrella Academy is that Ben, despite being a ghost that usually only Klaus can see, has also been de-aged. It's unclear what exactly the physics of ghosts are in The Umbrella Academy's world, which is a blend of science fiction and fantasy, but the fact that we see Ben as a child means it's possible his death may have somehow been undone. This is just a theory, of course, but it would be interesting to see Ben fully back on the team in season 2 - not to mention how useful it is to have his powers on the team.
The bigger question, however, is exactly what time period the Umbrella Academy time-travelled to. One possibility is that they could have simply gone back eight days again, and will have a third chance to prevent the apocalypse... though that could risk season 2 feeling like too much of a retread of season 1. What may be more likely, given the season finale's cold open, is that they will travel back in time to 1928 and meet Sir Reginald as a younger man. That way the audience can learn more about the backstory of this enigmatic character, and the Umbrella Academy could avert the apocalypse by saving Vanya from her miserable childhood. No matter where they end up, the fact that they've seemingly all been turned into children is probably going to throw a wrench in the works.
There's also the question of where Hazel and Agnes ended up after jumping away using the briefcase. It's possible that we've seen the last of these two characters, and that Hazel will enjoy a happy retirement of bird-watching and donut-eating with the love of his life. Then again, he could also prove to be a valuable ally if he ends up traveling to the same time period as the Umbrella Academy.