In order to get the quest, you have to go to the location shown in the picture above, which is quite far into the ocean from the starting place of the game. Make sure that you have a character that can deal wind elemental damage too. This is necessary for the quest later. There are multiple ways to get there:
The solo way is to go to Starfell Valley in Mondstadt and use Kaeya's elemental skill to freeze the water and create a path little by little. You can use the main character's Anemo elemental skill to help with the bridge making as well. Chongyu can work too if you happen to get him from the summons.
The multiplayer way is to have multiple people using Kaeya to form a bridge. You have to be at adventurer rank 16 to unlock the multiplayer.
Use the unintended(?) hidden route.
Glide to the island
The third optional way is to go to the edge of Starsnatch Cliff and glide over there. BUT! This requires A LOT of stamina to be able to reach it. You can try using Amber in your party to reduce gliding stamina by 20%, and also food buff that decrease gliding stamina further. It won't hurt to try it since, even if you fail, you will just end up in the last spot you were in before you started gliding. This method is also much much faster compared to the bridge making.
To reach the Stormbearer Point's island through the hidden route, go to the place shown on the map above and jump over the edge of the cliff
Near the bottom, you will see a small cave like opening. Go inside until you go past the swimming point.
Once you're inside the cave, simply head out of the cave and you will be able to walk on the floor of the ocean. Now all you have to do is just run all the way to the island. Be sure not to keep jumping, or walk on areas that will increase your elevation, or you'll end up back to swimming and have to start over.
Observe the shadow on the sundial
Once you've reached the area, head to the right side and start inspecting everything there.
You then have to use the "time" feature in the game to turn the time to 2 AM. You can use the wait feature by going to the main menu and selecting the "time", which is just below the mailbox icon.
Glide to the back area of the sundial and use the Elemental Sight (middle mouse button on pc, or tap the eye icon at the top left near the minimap) to see some swirling wind ball. Use an Anemo (wind) elemental attack on it to destroy it. Side note, once you're there, make sure to pick up the Anemoculus at the back of the sundial, as well as the luxurious treasure below the sundial and a little to the left, near the broken bridge.
Disperse the cluster of wind (0/3)
You will then be asked to find 3 wind orbs that you can see when you use the elemental sight. If they suddenly disappear, that's because the current game time is past what it needed to be. Simply use the wait feature again, and turn the time back to 2 AM.
The first wind orb is at the top of the tower to the right of the sundial.
The second one is just south from the first spot, over the broken down pillar. You'll have to use Kaeya again to make a bridge to get to the wind orb, unless you have Venti, then you can use a charge shot at the wind orb to destroy it from afar.
Last one is on the other side of the area, near a ruin guard, just above the rock.
Scatter the deposit of wind
Once you are done with all 3, head to the middle of the area and destroy the giant wind ball where you will then fight the Eye of the Storm. It is recommended to use some food buffs since he has high hp, and also don't use multiple wind type characters since it is immune to wind damage.
Pursue the Eye of the Storm
When it gets to low health, it will escape to a different area. A wind current will show in the middle of the area, allowing you to fly up so you can pursue it. You can also get there faster by teleporting to the temple of the wolf. You might end up missing some dialogue though.
Search for the Eye of the Storm
Similar to what you had to do before, once you reach the area, turn the time to 2 AM and destroy the swirling wind balls close to the sundial. This time though, there will be some slimes spawning here and there. It's recommended to take them out before you start the boss fight or you will end up getting overwhelmed.
Defeat the Eye of the Storm
The boss is more or less the same as the previous fight, except sometimes there will be more slimes appearing during the fight. Better to clear them out whenever they show up before things get too difficult. If the fight is too difficult for you, you can teleport away to the nearest location, which wouldn't be that bad since you can get to that location fast now, instead of having to make the ice bridge into the ocean again. Try it with multiplayer if things are too tough for you, or level your equipments.
Talk to Henry Morton
Once you finish the fight, talk to the npc near the sundial to finish the quest and get your rewards. You can't talk to him during multiplayer though.