I'd fly with my bike. As long as you pack it well you shouldn't have any problems with damage. The costs will depend on your airline, but I can't see shipping it separately being any cheaper. When you get to NZ it will be easier to deal with customs/immigration once for you and your bike rather than having to deal with the process at the airport and then collect your bike and deal with customs again.
There are no guarantees of course you bike can be damaged or lost no matter how much care you take when flying, but the same is true of shipping your bike.