I have study visa (Italy) which is valid for half year more. I am done with my studies and got an offer to work in Switzerland for 120 days, internship. Do I need a work visa to work there? If I need that what should be the correct procedure, as I am already in Schengen region, with student visa.
In the Schengen area, “short” means less than 90 days. Your Italian student visa does allow you to visit Switzerland for up to 90 days but not to work there. To the extent that an internship is considered work in Switzerland (it's likely to be but the rules differ from one country to the next), you would therefore need a separate work authorization/permit even if for a short internship.
But since your internship is longer than 90 days, it counts as a long stay and you definitely cannot use your Italian visa to cover your stay in Switzerland. You need another visa/residence permit in any case. You can apply for it through the Swiss embassy/consulates in Italy.