Most Asian cultures celebrate their New Year in the spring. This coincided with the use of traditional lunar months for the most part. It's not an accident that even Christian Easter happens when things are blooming. (Although it is weird, it is still being observed on a lunar basis.)
I propose the time around the spring equinox would make a perfect beginning of the year. After all, a new calendar is all about birth and renewal. Apparently, there have been a host of differing New Year choices in the past. I propose we start using April 1 for the our celebration.
Let's forgot about the ancient Roman God of Janus; Caesar was just exerting Roman power with that selection. We can keep all the months with their strange number of days. It'd be much too hard getting used to changes there!Spring: April 1 to June 30
Summer: July 1 to September 30
Autumn: October 1 to December 31
Winter: January 1 to March 31