Probably not, scince Signal doesn't really support secure web communication (without phone routing).

This phone routing story keeps getting repeated here but it's not true and never was. The devices share a key but are completely independent. You can easily try it, put your phone into airplane mode and Signal-Desktop works just the same.

Issue may not be the routing, but that a mobile device is required to create an account.

Are you saying that the Signal Protocol created Open Whisper Systems, which is lead by Moxie Marlinspike, does not require a phone number to work? (Say the Signal Protocol instead of the browser extension since the browser extension uses the protocol to work.)

As far as I'm aware the protocol can use other identifiers. If I remember correctly the development server allows registering without a phone number for easier testing of Signal-Desktop. Not sure what is used as an identifier instead. Not sure where I read it, so it may be the result of the telephone game. Take this with a grain of salt.

Signal also doesn't have a true desktop application. I don't want to install Chrome just to use a messenger.

What do you mean? Are you objecting to the phone number used as username?

It's not just a phone number, but also the IMEI. And the objection is that the service makes it (intential) very hard to use it anonymously.