Both my parents' home, where I used to live as a child, and the residence where I lived in my first year of university, seemed to have a "ghost cat." Like Kai Bengals, I often had the experience of feeling a cat jump onto my bed (usually down by my feet) when there wasn't anything there, or I would hear gentle footsteps in the hallway when I was alone in the house and knew for a fact that my parents' cat was sleeping in another room. Now, I didn't start noticing this at my parents' house until we had a living cat of our own, so there's this part of me that thinks maybe I'm just used to the sensation of cats leaping up onto my bed or the sound of them walking about in the house, and so I imagine it when they're not actually there?
However ... my cat Spike has developed a keen interest in meowing at a certain spot in our living room. He will leap up onto the arm of the loveseat and stare at the top-most hinge of a closet door. After a few seconds of intense staring, he'll start to meow; usually, it will be his anxious-alert meow (which can mean anything from "I want crunchies now!" to "I'm lonely!" to "Uh-oh, Mom has a migraine!"), and he'll keep it up for a good long time if you don't shoo him away. (And because it's kind of weird and funny, we usually don't try to stop him.) We've lived in this apartment since the beginning of July, and when we mentioned Spike's "invisible friend" to a friend of ours who also used to live in this apartment with her three cats, she got a kind of bemused expression on her face. It turns out that one of her cats also had a habit of staring at the top-most hinge of the living room closet, and would eventually start yowling at it. (Now, her cat also caught a mouse inside that closet, so that might explain why he was yowling at the closet door--but not why Spike would be doing it, too. The two cats have never met.)
Then, early this morning (like, around 2am or so), I woke up to go to the bathroom. Both cats were sleeping on top of me, and Spike got up to follow me. But once we got to the bedroom door, he wouldn't leave the room. He sat on the carpet and stared at me, with this extremely worried expression on his furry little face, as if he was afraid I wouldn't come back or that there was something in the hallway that posed a threat to both of us. I went to the bathroom, sans cats (Oz didn't even deign to get up off the bed, but this was laziness, not fear of the unknown!), and when I came back, Spike was still sitting in the doorway of our bedroom. He followed me back to bed, but stayed alert for quite some time afterwards. Very strange indeed!