Hi Yves and welcome to New Zealand.
Based on my experience with my wife - a recent immigrant from an Asian country - ANZ would be the bank I would suggest for a new arrival.
They are a registered bank in a lot of countries and probably because of this seem to be less afflicted by the narrow view of the world displayed by the more local banks.
Having a work visa will greatly help because it establishes you have a reason to have a bank account. Coming from a 1st world European country will also go in your favour.
Contact your bank of choice and ask to speak to their migrant banking division. They should be able to open an account for deposits only pending full validation which must be done in person at a branch.
When you arrive go into the branch with your passport and proof of address and hopefully that will suffice. Ask them at the time of setting up the account what is required.
One other thought - don’t use T/T (Swift) for transferring money if you can avoid it because they will likely bilk you in Switzerland and certainly bilk you here, and maybe even inbetween if you are unlucky. Instead find an independent remittance company who can do direct and fixed rate CHF -> NZD transfers.