I have SP a lot when I nap on the couch. I’ll dream that hubby, or fictional people, are walking through the front door and talking to me or shaking my shoulder.
When hubby was in Iraq and I was in the house by myself I had a re-occurring nightmares that someone was breaking into the house, night after night after night. It was always during SP that I thought I was getting out of bed and reaching into the bottom drawer of my dresser to grab the gun. Sometimes (in my dream) I would manage to fire off a couple rounds, getting the bad guy and going back to sleep. Other times, the magazine would be empty and all the bullets would be rolling around the bottom of the door and I’d have to fish them out. Sometimes I’d start having the weirdest conversations with my fictional assailants. It felt very, very real, and I could swear I could really and truly hear them talk, or feel them grab me, but I always knew it was a dream afterwards.
After a while, I would realize in my dream I would be dreaming and I would wake myself up. Other times, I would pretend I had magical superpowers and blow my attackers away with a flick of the wrist. Then I’d go fly off through the air somewhere.
What you're describing there is called lucid dreaming. You can actually become a full-time lucid dreamer, who embarks on fantastical adventures that seem completely real every night. I've tried to do it myself but the results are marginal at best.