Family history is that my mother was born in an air raid shelter in Germany on 4 September 1942. Is there a way to find out if this might be possible? Is there a comprehensive list of Allied air raids in WW2?
The 8th Air Force Historical Society maintains a list of all missions and related errata on timelines at their website here (under the "History" tab) There was no mission flown to a target by the 8th Air Force on that date, only the movement of the 352nd Bombardment Squadron (301st Bombardment Group) from RAF Podington to RAF Chelveston. Due to there being no 8th Air Force mission that day, but still an air raid, it is possible the mission you seek was flown by RAF Bomber Command; paging u/Bigglesworth_. If it turns out there was no raid at all on that date, it is possible your mother's birth could have taken place during a false alarm or something of the sort.
Do you by any chance know the name of the city your mother was born in?
The book The Bomber Command War Diaries has a day to day accounting of each operation taken by Bomber Command during the war. I don't think there's an equivalent online version.
On September 4, it lists:
A light raid of 6 Mosquitos in the area of Essen, Cologne and Munster.
A heavy raid of 251 bombers (a mixture of Wellingtons, Lancasters, Halifaxes and Sterlings) on Bremen. The notable aspect of the raid was a new 3 part tactic by Pathfinder aircraft to mark the target: "illuminators" dropped white flares around the general area, "visual markers" dropped coloured flairs on the aim points, followed up by "backers-up" who dropped incendiary bombs on the aim points. 12 bombers were lost on the raid. On the ground 124 people were killed and 470 wounded.