I generally tend to disagree with the "Dockerize Everything" crowd, but process isolation is one of the cases where Docker shines. And while it won't magically protect your identity, it's a fast way of having a browser start from a clean slate on each boot.

There's a lot of discussion on privacy-related Chrome issues recently on HN. Chrome/Chromium are excellent pieces of software, that are crucial to some of our workflow. If you ever need to run something like this, Jessie Frazelle's GitHub [1] has an excellent, evolving collection of Dockerfiles, for a wide array of software, that can be used as a cheatsheet.

[1] https://github.com/jessfraz/dockerfiles