The French tried doing something like this back in the days of the Revolution, dating things by the Era of Liberty, but it didn't really catch on.

So long as the years stay the same there's a built in reference to Christianity - and anyway I'd imagine that most people hearing the relatively unfamiliar CE and BCE would assume that meant "Christian Era" and "Before Christian Era".

I suspect that BC and AD will be with us for a long time. So will other dating systems, such as the Hebrew one - Happy New Year for the year 5572 on September 29th. (It's 1432 AH in the Islamic Calendar - and and 5111 in the Hindu Calendar...)