It's it possible to have a character moved from one content ID on an SE account to anther content ID on the same SE account?

Basically I have an alt that I really regret not setting up on it's own content ID so I could have them both characters logged in at the same time. Some of the content has me standing around for 10 - 15 min waiting for something to pop, and I wish I could level the other character while waiting.

I realize I am probably asking for something unreasonable, but I figured it can't hurt to ask.