The compass app may not work on your iPhone as it may get stuck on "Complete the circle to calibrate" whenever you open up the compass...
So here's a simple solution issue to the problem (Note: The instructions below were done on an iPhone 4S).
Basically, the steps to perform are these:
1. Open to the compass menu in the settings app.
2. Slide the "Use True North" toggle off
3. Quit the compass app from the multi-task bar
4. Calibrate your phone by spinning it around (it shouldn't be stuck now)
4. Enjoy testing and using the new revamped Compass UI