Actually this is one of the few reasons I do use Chrome on the desktop!
Chrome supports persona's so simply create a second persona and associate it with the other account. Now you can be logged into both accounts at the same time and still keep them separate. To switch, open a new window and click on the small rectangle in the top-right of the window with your name in it.
It is Microsoft I have this problem with since I have around 5 or 6 different accounts with them for different purposes.
For other browsers, you need to use a new Private Window but then you have to log in each time - Chrome keeps your Google accounts for you. The use of something like LastPass gets round the login issue by putting all of your logins just a couple of clicks away.