I've been using an independent calendar for my own records for over a decade now - it's the day-of-year, written base 7. It fits in with weeks; if we reset weeks every year (why not? we do worse things already with leap days and seconds) then dates become equivalent with weekdays. One number expresses the day-of-year, day-of-week, week-of-year, and month-of-year, using a 49-day month.
I don't really expect many other people to use this, despite being a better designed system. I think hardcore math/cs people would like it, though. I've written some small pieces of code that work with this system, if anyone's interested.
The 7date doesn't say much about how we count years. It's all about how you count days and other sub-year units. You could combine it with any method of counting years that you like. I defaulted to using the year system I'm used to because it's convenient.