The Deepshore Fossil is an item that is obtained from Layer 2. It is a biomechanical organism used to create any non-exclusive Enchant Stone of your choice. It will be consumed on usage.
Quest

Klaris Llfiend, Mother of Lights at Castle Light in the First Layer of the Depths will give you the Fossil upon telling her information about the Second Layer. This quest is not repeatable.

You will be able to tell her this once you meet the following requirements:

• Have obtained Diver Journal I, II and III from the corpses in the Second Layer:
  • The first in the left wall of the Nest Corridor
  • The second on a rock surrounded by Carbuncles somewhat near the Union Hook to Pilgrim's Respite
  • The third on the body of a deceased Diver immediately off the bridge and to the left once entering the Ethironal Shrine Sanctuary
    • Note that the Journal items themselves are not necessary, what matters is if you picked them up from the corpse.
• Have completed the Galewax quest at least once.
• Have beaten Chaser at least once.
• Have two Talents obtained within the Second Layer:
  • Ethiron Gaze, which is obtained after interacting with an ice spear inside the Firfire Caverns, and talking to the Leader located at Firfire.
  • Union Hook, which is obtained after going to the powered Frontier Furnace and speaking with Alpha, or obtained innately by the Ignition Delver Origin.

You do not need to accomplish all of these requirements in the same run or even in the same order, nor do you need to complete the current run to get the requirement (save for the Chaser clear).