Here’s a thing. Sometime back when the like button on Twitter was a favourite button, people used to favourite tweets as they happen for reference in the future. Now that’s not the most likely thing you will do if you are stalking someone. For the record, I do not endorse stalking people on their Twitter timelines, but it does happen.
For those wishing to be alerted every time someone or account tweets, you have the option of turning on mobile notifications, from their profile. But you need to be following them to do that. The new way to do it, (not quite new) if at all you already use Telegram, is by creating a bot via @botfather and setting the command to alert you of every tweet from a specific Twitter account you will set during the bot configuration. Remember the rise of the bots conversation?
Now what this does is give you a silent stalker status and also have a backup of tweets from the account. This means that when said account being 'monitored deletes a tweet you will have a copy saved on your Telegram app.
This information is not only good for serial stalkers, it’s good for others to know that there is this other way to be stalked outside of tweets being favourited. Another thing, if you favourite tweets they appear on your followers’ timeline as suggested tweets. So they will know what you have been up to.