I purchased Adobe After Effects downloading it to my laptop and was wondering if I did so with another laptop computer if the program could be operational on both computers at the same time, sharing the same serial number. If not would that mean I'd have to purchase another, or is there a different solution?
That sounds like a question for the Adobe team and not a user-to-user forum. From what it sounds like you're doing, I would tell you that you should purchase another seat, since the point of selling software is to sell it for each instance of it being used. However, I'm not Adobe and their policies may be different than what I think.