I'm watching the Funimation dub of Shin Chan, and have come across the various episodes about "Happiness Bunny's Revenge", where Nene / Penny's rabbit doll comes alive whenever the moon is red.
The episodes always give me the creeps and seem really out of place for a children's TV show, as it often depicts the rabbit with glowing red eyes chasing after Nene / Penny or just being unpleasant towards the other characters.
I suspect these are nightmares she's having as a result of feeling bad about hitting the rabbit when she's angry, but some more info about the original Japanese story / stories would be fantastic.