In order to get all the five purple gems you first must find "Booster the Explorer" which is the "white demon map marker" located in Cozy Cove center of the map. From there you must go in the direction that the telescope is pointing. After that you will find a second "Booster the Explorer". You keep doing this until you eventually reach a purple gem hovering on the water. Do this 5 times and you have completed the mission for all 5 purple gems.