"Madotsuki's Ghost" is a translucent version of Madotsuki which can be found in the center of a room very similar to the giant, red maze, Hell. The ghost can be passed through, but it can only be interacted with the Cat ability. The ghost always faces the direction the real Madotsuki is in.
Finding Madotsuki's Ghost
From the Number World, go through the door at the far left into the room full of beds and cupboards. Go down until you see a Toriningen and stab her with the knife, turning her hostile. Quickly go to the left towards the row of cupboards and go into the third cupboard from the left. This will transport you to a different, smaller version of Hell which is much less confusing. Madotsuki's Ghost will be standing in the center of the map. If you have the Stoplight effect, then you can also freeze the Toriningen in place after stabbing her, to give you a bit more time.
Trivia
Madotsuki's Ghost is a faded out version of Grayscale Madotsuki.
It is said by some that Madotsuki's Ghost is a referance to Survivor Guilt.