I suspect this might require a 3rd party application.
There is no way that I would want SMS notifications to my phone (which is almost what I'm looking for).
And if I could get emails just for particular users that would be even better.
You could try TweetyMail.com. From their FAQ:
tweetymail is a complete email interface to Twitter.
You could use a RSS-to-email gateway like feedmyinbox on the RSS feed for the twitter timeline in question.