This is not a request, but an order.
I'm on my second playthrough in Skyrim.
I love these types of exploration games, no matter how shitty the storyline may be. Nevertheless on my first playthrough I always automatically stormed straight to the green/red arrow without a second thought, I experimented turning it off in my second one and I feel sort of free.
I will be looking through every crook and cranny looking for whatever object/person/enemy my devout quest journal sends me to, taking in all the details and taking in way more of the game then I did before. Considering I played nearly all the quests on an earlier playthrough it's remarkable how little I remember about the actual context of the quest while running from one arrow to the next. Now I have to read all the books and journals I find if I want to know where to go/what to do next with no little arrow helper holding my hand. Now I also stumble upon most of the little out-of-the-way details Bethesda developers like to put here and there.
So turn off your compass marker.
Post script: After I found out how remarkable change it was, I also turned off the not-yet-found-location-nearby sense the Dragonborn somehow possesses. That also improves the exploration feel of this game a lot.