You sign in to your Google account in Chrome, then either there or when you go to your GMail account a message pops up asking if you want to use desktop notifications. After that you don't need the GMail tab open. I think you still need Chrome open, though.
Using Chrome and GMail with desktop notifications turned on, or with a separate email client e.g. Thunderbird. Not sure if IE and outlook.com work together nicely.
You sign in to your Google account in Chrome, then either there or when you go to your GMail account a message pops up asking if you want to use desktop notifications. After that you don't need the GMail tab open. I think you still need Chrome open, though.