How to get out of Exodus:
Note - You must have Jetpack
After Chekpoint Bravo, you will take a journey on Falcons and end up on a beach with two turrets, that you must turn on to end the mission. Do not turn on these turrets. Instead, look out to the bay. You should see a city on the other side. Follow the shore going right. You may pass a wraith on the way. Once at the end of the beach, on the right side, you must jetpack up onto the rocks above. Do not jetpack to the highest rock. Jetpack to the second highest rock. Now restore your jetpack. Saving as much fuel as you can, jetpack up to the tallest rock, and quickly made your way to the edge, and jetpack over the water. You want to do this quickly because there is a death barrier. You have to jetpack past the water, onto the next rocks. Now, you are free to go anywhere.
To reach the purple room, keep following the shore. You should eventually hit a barrier. Once you hit that barrier, look the opposite way of the bay, and you should see a building. Jetpack up the the building. Once in it, while on the building, start walking the direction you came (where the turrets are). While still on the building, you should see a room. Go into the room. Stand next to the door, and have your buddy kill you. Congrats, now whenever you are near that door, you turn purple.