Q: iphone 6 purchased in usa does it work in india

Depending on model, depending on NOT being locked to a US Carrier

You will have NO warranty OR support in or out of warranty in India on a US purchased iPhone

The warranty and POST warranty are  NOT International and is only available in the Country of purchase

Buy in India it will likely be the cheaper option in the long run

If the phone is unlocked, it surely will work worldwide. iPhone 6 are all GSM capable so just make sure the phone is unlocked or if you're getting it locked to one carrier, you've to get it unlocked. Check for AT&T or T-Mobile iPhone 6 which are GSM only as doesn't affect reception after unlocking.

The answer depends on what you mean by "work worldwide".  If you are referring to voice calls and SMS text messages, then Yes it is very likely it will work worldwide if unlocked to be used on all carriers.  However, the internet connection at higher speeds like 4G and LTE will only work if the local carriers you want to use, are utilising the 4G bands supported by a particular iPhone model and are supported by Apple in the sense that their carrier updates can be installed on that particular model of iPhone.  The main difference between the different models of iPhone 6 is precisely the high speed bands supported by each.

I think the advice "if you are using your iPhone in India, buy your iPhone in India" is indeed a very sound one if you want your iPhone to work with all services and to get support for any issues that might arise.