It would be nice if we can have more robust mobile side push notification, and I don't think it's super complex to implement (I can probably finish it in a week myself). It's like:
1. Build a mobile login API, that accepts: username, password / fg password, verification code, device info, and a push token.
2. In the backend, associate the push token with user info.
3. When a browser push notification is delivered, also send a APNS or GCM push package. The implementation is everywhere with all backend languages.