I do not think that this would be that big of an idea, as in, it wouldn't be worth implementing.
Having a team mode really only makes sense if you have a way of impacting your teammate's run, for example a Portal-like progression, where certain players need to be on certain pressure plates, however I see more problems with that than I see benefits coming from it. It's a cute little idea to be truthful, but as mentioned earlier, I doubt that this is worth the time.
But still thanks for bringing up your idea, even if it doesn't always get the best feedback, it's still highly appreciated that you think about ways of improving the community and gameplay!