Unfortunately, you can not move the chat/message bubble. However, there is a smart trick which you can use to see what is beneath the bubble.
For Windows users:
Hold the CTRL key and use + (plus) key to zoom in on the screen or hold the CTRL key and - (minus) key to zoom out on the screen
For Mac users:
Hold COMMAND (⌘) key and use + (plus) key to zoom in on the screen or hold the COMMAND (⌘) key and - (minus) key to zoom out on the screen.
The combination of Zoom IN and OUT feature will allow you to see/click below the chat or message bubble.