For what it's worth, an employee of one Kantonalbank mentioned to me, unofficially, that they want to know that you really have some connection to their region/community before they will open an account. Without his coming right out and saying it, it sounds like they want to hear a plausible motive that has no smell of tax evasion.

Similarly, after filling out the online forms to open a PostFinance account, I received a questionnaire in the mail, asking things like the purpose of the account, the source of the funds to be deposited in it, etc., and telling me I must appear in person at a PostFinance office to make a certified copy of my passport before the account will be opened.

Question for any who have opened a PostFinance account: Did you walk out with a Maestro (or some other debit) card? (One of my purposes for opening an account is to be able to pay for things while I'm in CH without having to carry too much cash, and I'd like to know whether they give a card immediately, so you can begin paying from the new account without delay.)