cough. No, thats not how it works. You cant do anything with a baro alone, let alone without any sensors. Its basically impossible to fly a multirotor without at the very least gyro sensor stabilization to measure angular velocity. Gyro/acro/manual mode is the simplest form of stabilization where the craft does not know what is up or down, it doesnt know its orientation, it wont self level, it will just resist rotation if its not commanded to rotate. If you fly in a certain attitude, like 40 degree forward, it will maintain that with sticks neutral. To get back to level, you need to pull back on your pitch control stick until it is level. You can do flips, fly inverted etc in this mode. DJI has never implemented this AFAIK, and I would expect 95% of its customers to crash in this mode. It is the most used mode on racing quads and for aerobatics.
The next step up is adding accelerometer data (absolute orientation) and having the quad automatically hold a certain bank or roll angle that corresponds to your stick position. So if you center the stick, the quad autolevels. To fly forward you must maintain forward stick input. This mode is typically called attitude or angle mode, it requires neither a baro nor a compass, just gyro and accelerometer data.
A baro is optional in any flight mode and only allows altitude hold where the throttle stick controls altitude rather than motor power. Which again is the default for DJI. IF you add a compass, its usually called mag mode or course lock, and as someone mentioned, DJI also used to call it just that with the Naza. Its needed for GPS navigation and position hold modes, because GPS only tells you your position, but not the direction the quad is facing, so it wouldnt know in which direction to correct its position. Compass data is not needed for attitude mode on any other flight controller, including the DJI Naza.
Anyway, how about we settle this once and for all. Who is brave enough to test? Disabling the compass as someone mentioned is not entirely conclusive, what would be conclusive is someone taking off normally, hovering in attitude mode, and then putting a rare earth magnet right on the magnetometer. If the craft remains controllable its pretty obvious you do not need valid compass data. If it does go out of control, anyone who's ever lost a drone should join for a class action law suit against DJI.
If anyone is up for this challenge, please be careful.