You can run Chromium with an argument from the command line like this
chromium --user-data-dir=/foo/bar
This will enable you to run it as root. This is not advised though for security reasons.
If by running as a normal user you mean execute the command as someone else, you can just do
su - username -c "chromium --user-data-dir=/foo/bar"
if you need to use the user's settings, then the path(user-data-dir) should point to them.