Looks like Microsoft Bing team has recently added support for many Indian languages for Bing on the web. Below the search bar, users have option to select their language of choice. Once clicked, Bing home page will be presented in that language. Microsoft is also allowing users to enter their search queries in their own language using a virtual keyboard.
As the user types their search query, even the search suggestions will be displayed in their native language.
Microsoft Bing team is now working on bringing Cortana experience to Indian users and it will be launched soon.
You can check out this new Bing experience in India here.