How to create a system entailing four characters in a platforming sidescroller?

You could use the RPG-caterpillar method, but seen from the side. Basically, everyone would follow the same input as the lead character, but with a time delay. That way, if you jump over a gap, all the characters would appear to jump over it independently, but if you make it, they all do (and if you don't, they all fall in right where you did).

You could also make it seem more like the player's controlling a team instead of one character by giving each party member moves that they can use when not in the lead, like healing or attacking from the back. Team attacks would be easier in this format, too, and you could still have the AI take control in puzzle rooms that require the party to split up.