I have twitter installed on my iPhone and wanted to know if it was possible to have my phone alert me when a particular user tweets?
Yup, it's possible. Click on the account you want to get tweets notifications from. And click on the bell Icon
One way to get it done, without being dependent on Twitter's mobile app, is by using a free action-trigger web service like IFTTT (If This Then That). It allows you to set new tweets from any account of your choosing as trigger and set the action to be an e-mail, SMS or even a mobile alert.
Here is a link to a ready-made IFTTT recipe which will send you an e-mail each time a certain account of your choosing tweets a new tweet.
The caveat with this is that it is not really real-time but works on a 15 min. refresh cycle. That means IFTTT is checking for new tweets every 15 min. or so...