Enter the Bunker
After you've talked to Buck, enter the bunker. Inside, you will find ropes hanging loose down the wall, which you can use to climb higher.
Once on the top, go a bit to the left, and then right, where you will be able to jump over a large hole. Climb down the decrepit stairs to your right, and jump down. Swim through the tunnel behind the destroyed grate. You will finally reach an area patrolled by several pirates. I recommend that you now go ashore and get into a fight. After you kill out all of your enemies, pass through the tunnel on the left.
You will reach a cave that is guarded by further enemies. Switch sides to the right, where there, atop the elevation, is an enemy to kill quietly. At the cave's other end, there is a pirate at the stationary cannon that you need to kill. You can try and kill the guards at the detonator, and blow up the entrance, and the abovementioned guard, You can also go for a safer method, and trigger the fight by killing the gunner first.
Whatever you decide to do, blow up the rocks that block off the passage out and pass through the door. At the end of the path, you will have to jump into water. Get ready to put up a fight with a crocodile in QTE. Swim underwater over to the other side and get out of water..
Make your way to the treasury
Open the door and jump into the water again. Swim over to the left side of the room, and get out of water. If you try to jump over the hole in the ground, you will fall down.
Resurface now, and enter the room with doors on both sides. First, enter through the ones to the right. Smash the glowing supporting elements, in order to flood the room. Leave, underwater now, for the previous room and take the door on the left. When it jams, resurface and swim above the jammed door.
Smash another three supporting elements. Once the room is flooded, resurface and mount the wooden structure and pass over it to the other side.. Jump back into water and return to the central area. Water level is high enough now, for you to get out of water.
You will reach an area with an altar in its central part. Collect from it another part of the compass. Walk over to the left side of the area, where there is a narrow wooden passage (the screenshot), which you need to take, and press the button on your way.
Enter a narrow passage to get out of the cave and complete the mission.