Make a new shortcut to your diablo III.exe
Right click the shortcut and choose properties
In the "target" line add -launch after the quotes so it looks like this
"C:\Program Files (x86)\Diablo III - Copy\Diablo III.exe" -launch
this shortcut will now bypass the launcher and allow you to open more than once instance of the game at once
NOTE: If you run the two instances of diablo from the same install you may experience ridiculous load times and slowdown.
You can counter this by doing the following:
Restart your pc completely so that no files are in use.
Make an exact copy of your diablo III directory.
Run each instance of diablo from a seperate copy of the folder. The issue seems to come from running them from the same files. (I can only assume the files for gameplay are heavily compressed within the folders and it has a hard time when 2 instances of the game are using the same files)