OK, here's what else I could find . . . apparently that there were actually
four different 'Traumwagens' built by Borgward between 1955-58. The first version was designed by ex-Tatra employee Erich Übelacker and featured an all aluminium twin OHC flat-four driving the front wheels, and an aluminium body on a steel frame. It was badly damaged in a crash in August 1955, and the phase II car was built around the remains. This seems to be the one that is shown in your picture; Main changes were the lower fins, a different one piece roof and restyled tail section. This version of the car also had a Bosch fuel injection system and an 8 speed pre-selector gearbox (..!!) which resulted in the claimed (by Borgward) 125mph top speed..... It was finally scrapped in 1958. Two further 'Traumwagens' were built, but I'm not sure if they were totally new cars or evolutions of this one.
I'm not entirely sure what you're looking for in the "why was it built?" question. After all, why are
most experimental cars built?
One article (all in German, thanks for
that experiment in translation on my part!) quite strongly stressed Borgward's exploration of lightweight aluminum construction and aerodynamics - at least that's what I
think it said - but somehow I believe you've got something else in mind? Borgward did help pioneer the automatic transmission in passenger cars in postwar Germany - so perhaps that "8-speed pre-selector gearbox" in the Traumwagen was an early step down that road.
Another article (again, all in German – I wish I could translate it better) states that the man from Tatra, Erich Übelacker, wanted a test vehicle for a new generation of engines. He designed this “Dream Car” with front-wheel-drive, disc brakes of his own design, aluminum body construction and a huge hinged Plexiglas dome (with no windshield wipers!). Different boxer-style engines of 2 or 2.5 liters were fitted, both in carbureted and fuel-injected form. Übelacker apparently even entertained the idea of having the entire exterior of the car’s body gold anodized . . . and in fact there are 1/43-scale diecast replicas of the car available, some of which have this finish. After it’s 1955 accident (reportedly due to failure of the experimental disc brakes, with resultant intervention by a tree), the series-one car was rebuilt to the style you pictured. Although most sources say the car was scrapped in 1958, at least one witness said the car was still in the Borgward truck plant at Osterholz-Scharnhorst around that time. So who knows. maybe it’s still around somewhere.
Am I sneaking up on that point yet?