I am trying to install Illustrator CS 6 on 12.04. I know it is possible, because there are some screenshots and movies over internet showing that some of people succeeded. However I did not. I am willing to use even CrossOver if necessary. Any help would be appreciated.
I seriously do not think that you might be able to run any product from the Adobe Creative Suite 6 natively into linux.
Even to be able to run this kind of software, you will need to use software such as wine. But even this way, software like Adobe Creative suite 5 are known to have severe issues, so great as to render them unusuable at times, so I wouldn't bet on Adobe cs6 to run properly. [EDIT]: Actually this suggests that it is usable, marking it with a Silver rating, however I wouldn't take it's word for working professionally with it.
If I were in your shoes, I would keep a Windows machine, or (as @January suggested) if I couldn't spare one, I would use a virtual machine. Please keep in mind that a virtual machine will have a great performance hit when you compare it with a physical machine, especially when workloads involve heavy GPU usage.
[EDIT #2]: To be more accurate as why you will have a performance hit using a virtual machine, first of all you have to note that the virtual device emulating the GPU is never able to emulate the GPU on its full potential, meaning you only get a fragment of its power. Additionally, the virtual device can use only a percentage of the GPU's RAM so, this is not going to help you with applications that do heavy usage of the GPU's RAM.