Is... Anything new? ...in its short form a perfectly natural question to ask someone if you want to know if anything new has happened (in their lives maybe)? Is it enough to be understood?
This expression is an example of colloquial omission, which is quite informal. The complete sentence would be "Is anything new?".
Out of "What's new?" and "Anything new?", both are OK but the former is more widely used and is, in my opinion, the better choice.
"What's new?" is an open question, so the person can say anything they want in reply, where as "Anything new?" is a closed question, which encourages a "yes/no" answer.
Anything new? is a complete sentence and not an idiom.
It has to be taken in context though, because the subject "something" is not specific. However the topic of the question may also not be specific. Therefore the context is often dictated but the location as opposed to a conversation.
If you arrive at your desk and ask your assistant if there is anything new it is obviously a work related question, whilst if asked when returning to your seat in a stadium you are referring to the match or competition.
anything C.E.D. pronoun (SOMETHING); used in questions and negatives to mean "something":