ICAO stipulates that it needs to be surrendered to customs when leaving the country, and they forward it to whoever regulates Aviation in the country (ie. CASA or CAA etc). Turns into a pain in the arse for long-haul airline pilots.
Now seriously....just keep using whichever logbook you like. New Zealand hours count 1 for 1 here in Australia (many would argue), just like anywhere else in the world. It's a log of your flying experience, wherever in the world it may be.