In order to find the compass you need to buy a Key from the Mysterious Shop (Hot Air Balloon symbol in the map).
Once you buy one of the keys it will give you access to one of the Labyrinths (new location added to the map). Head to the Labyrinth and complete all Puzzles in the Labyrinth until you enter the last room where you can find one of the items. If you complete all 4 labyrinths one of them will have the compass.
Video showing how to obtain the items and also escape the islands: