I was interested in reading A Short History of Byzantium by John Julius Norwich, but it was released almost 20 years ago (1997) and I was wondering if it's considered to be dated, and if there were other more recent and complete books about the Byzantine empire.
You are right, Norwich's book is both dated and not particularly good at giving you an accurate idea of what Byzantium was like. This is however to be expected - it is pretty difficult to condense more than 1000 years of history into a readable book that also encompasses the latest developments in academia. It does at least gives you an idea of who was important in the empire and the basic details of who did what to who, but obviously in many cases what actually happened was very different to what the sources tell us. A surface reading of the sources is simply not enough, yet this was the approach taken by Norwich in his attempt to produce an accessible introduction to Byzantium.
I've been giving this some thought and I think Judith Herrin's Byzantium: The Surprising Life of a Medieval Empire (2007) is the easiest way to ease yourself into Byzantine history, as it is both recent and covers stuff on the cutting edge of history (there are some excellent chapters on women and art in Byzantium for instance). Averil Cameron (one of the biggest names in Byzantine studies) also wrote The Byzantines (2009) and it is very good as well. It is supposed to be an introductory text aimed at both non-specialists and Byzantinists, but I think Herrin's book is more readable and provides a more innovative approach to studying the empire. Still, give them both a try and hopefully one of these books will get you enthused about everything Byzantine!
I am not a historian or a scholar, I just like to read on things, so maybe this will help :
Lost to the West: The Forgotten Byzantine Empire That Rescued Western Civilization by Lars Brownworth.
It might be an easy introduction in to the subject. It's more accesible than special history books.