Several months ago my outside temperature suddenly displayed "-40 degrees" and my climate control stopped working properly, including a complete lack of air conditioning. Also once that the tank of fuel was refilled a few days later but it stayed on empty with the fuel light on.........both very annoying prblems.
I replaced the outside temperature sensor, but that didn't solve the problem.
I then bought a second hand instrument panel (not knowing that you're not suppose to change them) and fitted it.........BINGO!.. it cured it, but only for 1 week when it reverted back to the original two faults.
I took it to an independent BMW specialist(together with my old panel) who checked the fuel sender unit/wiring and said that it was functioning OK, but they couldn't cure the faults.
I have also took it to a BMW main dealer who spent an hour on it and couldn't find the fault and would me to authorise a further £300 of diagnostic cash.