Bushy,
CASA is specifically barred by its legislation from exercising any commercial regulation, any such powers were removed years ago.
Apache,
Matter generally between Australia and New Zealand are governed by the Trans Tasman Mutual Recognition Treaty, and legislation flowing from that Treaty in both countries.
Naturally, this includes a wide range of aviation matters, including conditional mutual recognition of AOCs.
An Australian base for any other "foreign" operator, and what rules apply, is entirely dependent on the facts in each case.
However, in principle, you can use an aircraft and crews of any ICAO state on an Australian AOC.
Don't ever confuse the bureaucratic complexity and cost of actually getting an Australia AOC with safety outcomes. Do any of you actually believe, as one example, that Virgin Blue is fundamentally "safer" than any other related company flying under the "extended" Virgin banner, because they have an Australia AOC, rather than UK/NZ/DE/US AOCs??
If you have a look at the last ICAO audit of Australia, you will find nothing to support that contention, that Australia is "better". Quite the contrary, in fact.
Tootle pip!!