This. You absolutely must buy ARR (base game / Starter Edition) for all platforms that you want to log in on at all. You don't have to buy the expansions for all platforms, but each platform keeps track of which expansions you have bought for that platform, and if you are missing expansions, you can only access areas and content that you have the expansions for on the platform you're playing on. If you log out in an area that you don't have access to on another platform, you can't log in on that other platform; so really, you have to buy the expansions on all platforms.