Ghost Browser might be what you are looking for. It is a Chrome-like browser with special Multi-Session Browsing.
While it does not open a new session for each new window, you can have as many sessions you want even in the same window. It has color coded tabs for each session.
You can download it from their site:
And this is how they describe this feature:
Say goodbye to the inefficient methods of using
browser profiles, multiple browsers or browser extensions with bad
interfaces just to log in to a web site with multiple different
The multilogin capabilities of Sessions are the solution you’ve needed
the whole time and the only one you’ll ever need again.
Sessions are color-coded tabs that have isolated cookie jars so you
can handle multilogin with ease – in one browser window. You can even
name the Sessions to help you stay even more organized!
Once you start using Sessions, you’ll wonder how you ever lived
If you want to use Chrome, there is a workaround but is isn't that easy to use.
You use the following command to open a new temp profile of Chrome which is completely a separate session:
But each time you will need to change the directory to be 2, 3, 4 etc.