There used to be an addon called MobileFrames. It allowed me to reposition standard UI windows (bags, character frame, trade frame, trade skill frame etc). Unfortunately, it stopped working on PTR 4.0 and I couldn't fix it myself.
Since this kind of functionality was important to me, I decided to write my own. Standard disclaimers apply: this is an early version, bugs happen.
Note: Versions 0.3 and later will try and prevent standard frames from opening outside of the visible screen where they can't be dragged. This can result in wonky initial placement. Simply drag the frames where they should be and everything will be alright.
The following frames have been made mobile so far:
All bags, including keyring and bank bags
Durability - new in 0.4
Casting bar - new in 0.4
Quest log detail
Game menu and all related windows
Class and profession trainers
The following windows are no longer affected by the addon:
Item link tooltip (it's movable without the addon)
World map (it behaves strangely, so the addon won't affect it until I figure out how to deal with it)
I am pretty sure I forgot a few windows, but they are easy to add as long as someone posts a request on the forum. *hint hint*