My first cat died on December 31, 2012. I'd had her for about 17 years, so I was very upset when she passed. She was my best friend. After she passed, I had several dreams in which she was alive and I was petting her, cuddling her, etc. I would wake up crying because I realized she was dead once the dream was over. It was tearing me apart.
In October 2013 I finally decided to adopt another cat. This particular one reminded me very much of my old cat, same markings but different color, and she was around the same age of my old cat when I first brought her home. New kitty was traumatized from being at the shelter so it took her a few weeks to adjust to us. In that time, I started noticing that even though she wasn't interacting with us very much, she was always playing by herself. And I swear I saw her chasing a "shadow cat" many times. This thing was not her own shadow, as it moved freely on its own. I would see it out of the corner of my eye and obviously the new cat saw it too.
One day, a few months after we'd had the new cat, I was in the kitchen and a cat walked in. I saw it out of the corner of my eye and assumed it was the new kitty. It was dark colored and the new kitty is dark. It rubbed up against my leg. As I stooped to pet it, nothing was there. I went into the other room and new kitty was asleep in the arm chair. Just judging by how comfortable she looked and how deeply asleep she was, she had been there for awhile. Whatever rubbed against my leg had cat hair because I felt it.
I've never felt afraid of it but stuff like that has happened many times since we got the new cat. I have other stories too if anyone is interested. My house seems to be a spot for weird happenings.