Your mileage may vary when you try to open an account.
Permit? American? Job? Raining outside? Bad hair day? etc etc
There are a few threads here on the problems folk have had.
Safest bet in Zurich is UBS at Glattbrugg or CS at Uetlihof.
They are used to foreigners wandering in and asking for bank accounts.
Otherwise, pick your nearest one (any bank) and try them...
Personally...
The Post Office were quite rude to SWMBO when she tried... she tried them because UBS had imposed restrictions.
UBS Kloten also argued with here about how our UK bank accounts worked
On the other hand, I had no problems at all with UBS Glattbrugg...