What research has been done on the effects of requiring students to learn to count and do some easy arithmetic in an alternative number base, for example binary, base four, base six, base eight, base twelve, base sixteen, or base twenty?
I can't answer the OP's questions, but I'll just mention that a local 6th-grade teacher (in the U.S.) has a successful unit on base-$5$. It is mentioned in the recent article below. Sometimes he called it "star-fish math."
- James Henle. "Math for Grades 1 to 5 Should Be Art." Mathematical Intelligencer. 42, pages 64–69, Dec. 2020. Springer link.
"And then there is numeration. Kids can invent a method of naming numbers.
[This teacher] used to tell his sixth-graders every year the tale of the mysterious mathematician Professor Étoile, who invented a new way of notating numbers using the letters $A$, $B$, $C$, $D$, and $O$. His method was lost, [the teacher] explains, when he fell off a boat and was eaten by an octopus.
[The teacher] then divided the class into groups. Each group was tasked with creating its own system of numeration using $A$, $B$, $C$, $D$, and $O$. After a week, they presented their methods in a mini-math conference. [...]
[He] did this every year with great success. His students came up with many creative systems."
Of course étoile is French for: star!
I'd say yes, and I'd go with binary if you had to do any one alternative base simply because it's so relevant to computing and technology, and in my experience teaching discrete math, once you understand binary, related bases like octal and hex are pretty simple to pick up. But I don't think the converse is necessarily true.
Ideally I'd like math and CS students to learn base 2, 8, and 16, and then learn how base 10 notation is related. From there students can do whatever they want; I sometimes teach student base 36 and use 0-9 plus A-Z. After they learn the general concept of base-$n$ representation, it's all the same.
Using some familiar everyday objects to represent bits can be fun for teaching binary. For example, asking students how they might build a binary clock like the one below can generate some good ideas:
Or, ask students how they would do this if all they had was a single flag that they could either hold up or put down. Or my personal favorite, counting to 31 using only the fingers of one hand (literally, "digits").
For hexadecimals, this scene from The Martian is really good.
Finally, should language departments teach this? Heck no! Why should they get all the fun?