While that appears to be a viable workaround, I assume it would not only require more or less the same amount of work, but also would be harder to implement, since most users have different hardware / software / network configurations at their end. Apart from that, there would be the unnecessary need for the user's computer to be in Always On mode.
Wait, what am I saying? The above is not even an option. The way Push Notifications work, the server is first connected directly to Apple's own Push notification cloud that then pushes notifications via the API to the app. Of course there's more technicality to this, I need to read through the 3.0 documentation, but from what I understand, the app, when being used, interacts directly with the source data, in this case, twitter. Now, when you quit the app, the direct link breaks but Apple maintains a persistent link with their notification cloud and the phone. Hence, unless the source data (twitter), sends data to Apple, you won't receive any notifications.
