[Offshore]

Swissquote is a brokerage company, not a bank. You might try a debit card offshore and use it as your bank account. Fund the card via bank wire.

Can I open a Swiss bank account entirely via the Internet?

No, because technical and legal reasons prevent the customer identification procedure from being carried out entirely online via the Internet. At the present time banks in Switzerland must follow the identification procedures laid down for opening an account by correspondence. In accordance with the Due Diligence Agreement (CDB 08), the bank verifies the identity of the contracting partner by obtaining a certified copy of an official identification document (passport, identity card, etc.). The certified copy may be provided by a branch, representative office or group company of the bank; by a correspondent bank; by a financial intermediary specifically appointed by the bank; or by a notary public or public office that customarily issues such authentications. The bank also checks the address of the new customer through an exchange of correspondence. Certain Countries only apply for foreign customers.
There is no such thing as an "anonymous" account in Switzerland. Under Swiss law, the bank must know who you are.