In the 1960s life was simple. There was only one doctor costume available, and if you didn't like it, you could go pound sand.

Here we have Superman sporting the doctor costume:

In another comic, Bruce Wayne:

In yet another comic, one of Luthor's men:

And, finally, some other criminal wears the same get-up in another comic:

Does this costume fool anyone? Everyone owns one. And can you blame them? That is an excellent beard and moustache combo. Bruce Wayne really made it his own by switching up the lab coat for medical scrubs (PS - the nurse is Supergirl, and I thought she was drinking a soda, but I guess that's a roll of gauze).

And since we are talking about that panel, it's also a great example of how Superman solves simple problems with needlessly complicated solutions. Did he really need to bother Batman for this? Even better is in the next panel when the "doctor" tells Clark in front of Lois that he needs to keep the bandages on for two weeks. I think that was Bruce's pay-back for being dragged out during daylight hours to be a part of this absurdity.
Also...if Superman is so worried about getting those bandages on in a hurry, why doesn't he do it himself in a split second before Lois arrives?

Eventually I'll stop talking about Showcase Presents Superman. But who can really resist this stuff?