Android: Falcon Pro has been our choice as the best Twitter client for a while, but Twitter hasn't made it easy. After some trouble with authentication tokens, Falcon Pro was removed from the Play Store. It's back now, with a vengeance -- or at least in-app purchases.
The new version contains multi-account support which was around before, but many users weren't able to try it out due to the authentication problem. Now you have a second chance. It also has a slick new UI, updated for Android 5.0.
The free version is a limited trial, but you can pay $4.90 to unlock the full app. There's also a $2.45 charge per extra account added, which is disappointing, but presumably being partially used as a deterrent to users who would claim tokens and then never use them again.