Dungeon Doom Solution
Thanks to Paulon Dragon and Trigon Dragon for the maps.
The dungeon begins in Room 1. The room is surrounded with ethereal walls that can only be dispelled by the Sceptre of Lord British. The room has no exit ladder, so the only way to actually exit Doom is by casting Gate Travel.
If the Shadowlords are still alive, they will be waiting here to steal the sceptre, thus making escape impossible, and will then kill the party.
Every direction is the right one, but for this guide, dispel the walls on the west, and then follow the path to the ladder down, descending to the next lower level.
The up-and-down ladder in the middle might look tempting, but it is a trap leading to rooms full of enemies. All of the ladders that only lead down are safe, and it is your choice which one to use to descend to the lower level.
Search the eastern wall of the starting room on this level to find a secret door. Following the corridor leads to Room 2. Three daemons hide behind an ethereal wall, and the time it takes to dispel it with the sceptre can doom the party since they can Charm your party. Donning Lord British's Crown to negate their magic is needed to survive this room.
Be warned that removing the other two ethereal walls will reveal more daemons to fight.
From now on, keep the crown on to protect yourself against your foe's magics, and descend down the ladder in the room to the lower level.
Level 4 and 5
These two levels are interconnected in confusing ways, making it necessary to change between levels a few times.
Coming from level 3, there is no choice but to directly descend to level 5. On level 5, follow path 1 to the east, and then climb up the pit with the grapple to return to level 4. On level 4, follow path 2 to the south and climb down the first pit, then follow path 2 on level 5 first to the east, then to the north, and climb up the pit back to level 4.
Back on level 4, follow path 3 first east and then north, and then search to the east to reveal a hidden pit. Descend it and you reach the ladder down to level 6.
This level is simple and straight forward.
Follow the corridor until you reach a room in which you are greeted by numerous kraken and sea serpents. After you've gotten rid of your foes, follow the path to the south to reach the ladder to level 7.
Level 7 and 8
These two levels are the last and most difficult puzzle in Dungeon Doom.
Once you have reached level 7, do not descend further, as it is a dead end. Instead, go north to enter Room 3. The room is full of dragons, daemons and sand traps. The sand trap in the middle is concealing the ladder down (marked on screenshot).
This ladder directly leads to Room 4. It is possible to leave Room 3 after clearing it, resting, and coming back to go down to Room 4. In Room 3, one must remove Lord British's Crown after defeating the dragons and daemons to reveal wisps. Be careful, as they will possess your party should you remove the crown for too long.
Room 4 is filled with sea serpents and dragons. Kill as many as possible before shooting the NW wall (marked on screenshot) to create the bridge. However, this creates more monsters and a large swarm of mongbats, so one may want to flee the room with the Avatar first. As long as you can Resurrect your fallen, (the Avatar should be level 8) it should not be a large problem.
On level 8, follow the corridor and the ladder leads directly into Room 5. The poison field will vanish after some time, but that path is a trap. Instead, after killing your foes (some mongbats and reapers), push the eastern walls (as shown on the screenshot) of the room and leave south. In the corridor, search south to reveal a concealed pit. Climb down it to reach level 8, and directly go up the ladder you see back to level 7.
Carefully search the corridor on path 3. The first hidden pit is a decoy. After turning south, you find another hidden pit which is the right way. Descending it drops you directly into Room 6. The first important thing is to kill all the sharks and wisps. After they are killed, do not use the northern passage, as it is a trap. Instead, first stay on tile 1(screenshot), which opens a small alcove where you push the wall marked 2, to open the eastern exit.
Leaving the room, the ladder back up is directly in front of you. It leads directly into Room 7.
This is the final battle, and it is very difficult. After destroying the mongbats on your side, do not enter the square marked on the screenshot. It will trigger a horrible trap that floods the room with lava, spelling your doom. Instead weave the sceptre at the right place to open the ethereal wall and destroy your foes with strong spells while they try to fit through the small opening.
After leaving the room to the east, search to the east to find the last hidden pit. Now is a good time to save the game, before descending this last pit to Lord British's prison, Room 8. Walk everyone in front of the mirror, and then give Lord British his Sandalwood Box (say "yes") so that you can escape with the Orb of the Moons.