What's the reason that the alarm clock isn't working when the phone is turned off?
There is no power to the phone when it's off.  It's actually off.  Some phones have a minimalistic charging mode that just shows a battery animation, but I have yet to see an Android device that boots into a clock-only type mode.

The Android OS is designed to use minimal power when not in use.  If you have a problem with the battery dying overnight, it's probably due to a third-party app.