The number of 4 bedroom or 3 bedroom with den in Sun City West (Del Webb) with private pool are limited, but do exist in this Active Adult Community (55+). They sell quickly and are priced in the $300,000-900,000 range. Another nearby option is Sun City Grand (another Del Webb development) in Surprise. You are getting top of the line facilities in communities that have held their value better then most other areas.
Consider that most owners just use the community pool they pay for and avoid the maintenance of a private pool.
The link below shows 10 currently available properties that are 3 bedroom plus den and private pool.
The 4 bedroom homes mentioned in the other answers are NOT in the age restricted subdivisions of Sun City or Sun City West. The ones cited for Sun City are actually in the City of Peoria and the ones cited for Sun City West are actually in the Corte Bella subdivision.
I'm sorry but there are no 4 bedroom properties in any of the adult communities here since there is so little demand. Children under 19 are not allowed to reside in any of our age restricted communities and can only visit for 90 days a year. However, we do have some 3 bedroom homes that also have a den/office, if that would satisfy your needs.
You would be doing yourself a world of good by securing the services of an experienced, reputable Buyer's Agent/Realtor to assist you with your home search. Trulia is a great place to surf for general real estate information, but when it comes to finding specific properties, nothing beats the Arizona Regional MLS database. Your agent could program a customized/personalized home search and set up an MLS-direct portal for you that is based on your specific home search criteria. You will LOVE the system.