One-Punch Ma
R1: If Cole starts off alongside Saitama when he fights Crabante, he probably helps Saitama defeat the crabby monster. Cole heals Saitama's injuries and then goes on about his day to try and figure out what the hell is going on. Unbeknownst to Cole, his heroic actions and power inspire the aspiring salaryman, further increasing his determination to become the strongest hero, even if it means becoming nearly bald like Cole. Years later, the two meet again, Cole failing to recognize the bald hero, and Saitama instantly recognizing the nearly bald man who helped him all those years ago.
R2: Assuming Cole drops in before the Monster Association. Cole tries to live a normal life at first, probably going back to the bike courier skills, his powers and bravery helping him deliver packages where no other man will. His escapades and power eventually get the attention of the Hero Association, who send one of the S-Class to recruit him. Depending on who they send, Cole may turn down their offer, since he won't be impressed by an organization who would hire characters like Tornado, Pig God, Metal Knight, etc. If they send someone friendly (Bang, Darkshine, etc) Cole will be willing to hear the Hero Association out. Cole hesitantly chooses to join the group, wary of their shadyness, but interested in trying to help more people. This puts him in the crosshairs of Garou, the Monster Association, etc.
R3: Evil Cole runs around wreaking havoc. Hero, villain, monster, Cole strikes him down. He ends up living in Z-City, preferring to live in the monster ghost town, as the place reminds him of Empire City and New Marais, a place of no rules, where the strong are free to do as they please. He, alongside Garou, are identified as monsters by the Hero Association and ranked at level Demon. Cole's powers and general evil demeanor gets the attention of the Monster Association, who reach out to recruit him. Cole is wary at first, but the promise of more power lure him in. After meeting with the Cadres and learning of Monster Cells and the Monster Associations plans, Cole tells them he'll keep them in mind. After sensing something off about Gyro-Gyro, Cole decides to steer clear of the group. Of course, things aren't that simple, and he eventually gets dragged into the war between hero's and monsters, but serving as a wild card like Garou.
R4: Beast Cole kinda binds with Saitama with their sheer boredom with life. Though, they don't really get along, since Saitama criticizes his decision to wipe out humanity out of fear of sacrifice and the RFI not working, Cole criticizes Saitama's reasoning for becoming a hero, for fun, talking about how the greater good should come first.
It'd be an interesting relationship to say the least.