Here's one we bet most of you haven't stumbled upon during your search for that rare and unique bass - the Ampeg AEB-1, and it's fretless sibling, the AUB-1!
These Ampeg creations were the first traditional "horizontal" (not upright) basses of their own design, and they were dubbed "Scroll Basses" because of the scroll headstock shape. They were only made for about 2 years between Summer NAMM in July of 1966 until early 1968.
Obviously the first thing that'll catch your eye is the f-holes that are carved all the way through the body of this beast! Most of the top is also covered with a large black pickguard. It's an unmistakable design that isn't for everyone, but we think it falls perfectly on that line between weird and cool.
The AEB and AUB basses also used a dual-coil "mystery pickup" - as it was literally named in their catalog. It was placed inside body and under the bridge (no, not a piezo) and was designed so the bass could use gut strings - as the original designer of the AEB/AUB thought that fretted basses were a passing fad. (He was quite wrong!)
The Scroll bass also uses a unique tailpiece that requires "longer than long-scale" bass strings - in order to set a proper break angle over the bridge and pickup. The Scroll headstock is actually made from plastic laminates, though the neck is all maple.
If this unique Ampeg creation caught you eye, you're in luck because Eastwood has reissued it - in guitar form, as the Deerhoof Signature!