I like this idea. Unfortunately, it can't coexist with the Gregorian calendar in its current form; if a small fraction of the world switched to it, they'd always have to wonder which calendar someone writing "Jan 1" was referring to.
When a date is written down, it needs to be immediately obvious which calendar it's referring to. It needs a special notation that doesn't overlap with any of the currently-used notations for dates, so that it will be unambiguous (and so that the old calendar will remain unambiguous). I propose doing this by giving each month a letter, A-L, and writing dates like this:
2012-A01 = New calendar, first day of first month of 2012
2012-L30 = New calendar, thirtieth day of twelfth month of 2012
2012-01-01 = Gregorian calendar, first day of first month of 2012
C30 = New calendar, thirtieth day of third month of whatever year this is
With the notations neatly separated, you could then print calendars with the two schemes side by side, and let people convert back and forth for awhile while the new system is adopted. We could give the months fanciful names, too, as long as they start with the right letters. But the important thing is, the two schemes must be unambiguous and distinct.