According to University of British Columbia professor, Dr. Stanley Coren our fluffy friends do indeed dream like us! They could even be dreaming about their beloved owners.
An MIT study on mice discovered that animals enter REM sleep and observed brain activity continued whilst asleep. This backs up similar observations noted in sleeping dogs.
If you are curious to see their dreaming in action, you can observe their REM dreamlike state about 20-30 minutes after they have fallen asleep. Try not to disturb them though as they can wake up startled.
Just like when we are abruptly awoken from an amazing dream!
It although remains a mystery as to what our beloved pet actually dreams about.