The PDP-8/S was the serial model. Serial processors weren't uncommon in the "old days". Other examples are the F-14 CADC (an early MOS processor) and the Datapoint 2200 (whose architecture was turned into the 8008). Many of the early aerospace computers were serial because of weight constraints, such as the Arma Micro (on the Atlas), IBM ASC-15 (on the Saturn I and Titan) and the Autonetics D-37 (on the Minuteman).
And then there's the Motorola MC14500B, which was genuinely a 1-bit microprocessor. It was used for simple control operations to replace relay logic.