You need to have a passport to open a Swiss bank account.

In some cases we can open an account with an identity card. If you have a valid, official identity card, please fax a copy of it to ++41 21 331 48 40 with the type of account you want to open and your email or phone number. We'll tell you if this document is sufficient to open an account.

In all other cases the only solution is to get a passport. US clients can use www.passportexpress.com to get a passport in only a few days.