I think I learned that the cat first tries to ask for attention, then goes to sleep, then asks for more attention, then gets dirty and sad if left alone. I felt sad and responsible for it.
However, I feel like it could do more on its' own perhaps. Like, if it had toys it could play with on its' own, like a yarn where it bounces it around on a screen. Now, I haven't played any other idle screen-sitter games, so, I don't know how exactly it could work. For me, he is sitting on my second screen also.
Another thing I thought about is maybe it could also react to keyboard strokes somehow and follow the mouse cursor even while you are not interacting with it. Maybe. For me it seemed like the direct vs passive interaction ratio is too big on the direct.