Batman's first airplane appeared in the September, 1939, Detective Comics (no. 31). He called it his "Batgyro". It looked like a Etrich/ Rumpler Taube with wing planform modified to match the popular idea of a bat's wing. To make it a "-gyro," there was a four-bladed rotor on a structure right over the cockpit (with room enough for two side-by- side). That structure was much like the support for the rotor of the Pitcairn-Larson PA-39. The aircraft also has a single Vickers machine gun mounted on the centerline, firing through the propeller.
Has anyone ever seen this built as a model? Plans for both the Taube and the autogyro are available from Cleveland Model & Supply Company
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but these are for large wood models. I presume it's possible to find 1/48 or 1/72 kits to do build it out of plastic.