Maybe, maybe not. The specifics will depend on teh exaxct specs of the machine, what auxilairy hardware is involved and a million tiny things. CS4 was never tested or certified on Win 8. You won't know if it works until you try.
Thank you very much for your kind reply. One last question: If I buy the PC that I have in mind, and it turns out that it is not compatible with my CS 4, is there anything I can download or update from this page that would make it work?