You need to complete the following quests to get the Island Paradise achievement and enable you to hunt this gold key:
Gift Wood for Giftwood
Head to the main settlement and track down the flit switch. You need to hit with several attacks as indicated by its symbol. See the video below for a complete rundown on where to find the orb (flit switch) and the golden key.
Driftwood Gold Key
Enter the 3rd entrance of Chillbreath Caverns by checking this video:
Instead of taking the frozen waterfall path, continue taking the rocky path as it leads in the upper part of the cavern. Continue following the path and it will lead you to the exit. The Gold Key is located there.
Veiled Path (The Enigma)
Enter the Veiled Path by fast traveling to the Shifting Sands then continue to the left. The path is just a straightforward one just continue walking until you reach the long flight of stairs. The Enigma's door is the only one accessible here so its very hard to miss it.
Check the walkthrough below if you're accessing it the first time:
The Enigma (Veiled Path)
Grab the chest near the entrance. There's a ranged flit switch on top of the door so hit it to open the path. To open the individual rooms, you need to light up the unlit braziers so equip your Fireball gauntlet. Head to the next room and grab the Green Aurora Bloom in the middle, then open the room to your immediate left to find a chest inside.
In the next room, clear out the enemies and the doors will open automatically. Grab the Red Aurora in the middle of the room. The gnome is located in the middle door. Exit to the other door that leads to the third room where the Silver Key is located and some colored platforms.
Veiled Path Enigma
Resolve the puzzle by stepping on the following colored platforms:
yellow, blue, yellow, red, blue, red, blue, red
Enigma Platform Solution
Dispose of the two enemies that will appear then follow the long walkway until you reach the Gold Key. That's all there is in this area, so use that key in one of the four golden doors to receive your reward.
You need to purchase the ancient key from the sidequest “The Key to a Greater Key”, which will only become available after you beat the game (defeating the darkness).
Crossroads of Passing
From the stone arch in the Shifting Sands (there will be an outpost underneath if you accepted Kalin's proposal before) continue heading northwest until you find a small path leading to the locked door.
Once inside, you need to step on the lighted portions of the platforms to either raise the next platform or to spawn enemies. Your objective is to reach the two rooms on both sides of the chamber so you can step on the foot switches. These will light up the two braziers on the main door ahead. Behind that door is a Gold Key, the last one if you've been collecting them before you finish the game.