I was in Abu Dhabi and Dubai last year for a job interview. I checked out the cost of electronics pretty carefully, as had I taken the job, I wanted to see what I needed to buy in the States and what could wait until I got to the UAE. I found that most electronics stores had a smaller assortment of equipment than U.S. stores, and prices were 20%-40% higher than what I could get in the States. If you're planning on moving there, you want to take with you what you will need, because shipping charges will cancel out (and then some) any savings from buying through mail-order/online outlets.
There are excellent bargains on clothing and jewelry, especially gold, if you're in the market for those. I saw a Pierre Cardin store offering a complete men's business outfit (suit, shirt, tie, and topcoat) for roughly $200 (you have to convert from dhirams). You will also find that their shopping malls fairly sparkle.
One travel tip: there are services in both the Dubai and Abu Dhabi airports that will charge you roughly $50 to expedite your check-in and security processing, and then give you access to their first class lounge while you wait for your plane. Spend the $50. The public lounges are filled with expatriate workers going (or being sent) home, and the majority of them are laborer types (75% of the people living in the UAE are foreigners who are brought in to work for the Emirati. Any business operating in the Emirates must be at least 51% Emirati-owned. Nearly everyone works for an Emirati, and the Emirati appear to spend their days lounging in coffee shops). Personal hygiene doesn't seem to be a big priority. Trust me on this.