I see the green checkmark next to the other answer, but this fits way too well to ignore:
The transmogrifier is the "multiply by yourself" operation.
One said, "My little buddy should jump down, but there will be a new one."
Taking the name literally, "One" is 1, and his little buddy is -1. Were -1 to jump into the transmogrifier, you'd multiply -1 by itself, and there would be a new 1.
Jumpy said, "Why why, if I go in it 4 times I come back!"
Jumpy is the imaginary unit $i$. If he goes in four times, we get $i\times i\times i\times i\times i= i$, where the value jumps around the unit circle as the multiplications progress.
His brother said, "I do too!"
Jumpy's brother is $-i$, and $-i\times -i\times -i\times -i\times -i = -i$.
Jumpy said, "But you will have a negative, and I won't!"
$-i$ does, indeed, have a minus sign
Jumpy's brother said, "Oh, oh, oh, but you will become me!"
In the process of going around the unit circle and becoming themselves again, the brothers $i$ and $-i$ indeed temporarily become one another.
The Famous one argued, "You all are shameful. I go back to myself!"
That would be again the famous "one", the only member of the family (see below) that instantly becomes itself when multiplied by itself
His family said, "Well, I'm lucky too, we're really close, but we have an ugly multiplier next to us!"
The family is "roots of unity", where the four aforementioned brothers (1,-1,i,and-i) don't have any ugly multipliers preceding their real and imaginary parts, but the infinitely many other roots of unity do. They all live densely packed on the unit circle.
The Famous one said, "I don't! That's one!"
"one" seems have an inflated ego; While it's true that he doesn't have a multiplier, he seems to be forgetting that as a member of an infinite set, he makes up for exactly 0% of all the members. Maybe the fame of being the most famous number of all has gone to his head.
I went back to myself too if I went in, but people didn't care about me.
You were even missing from many early number systems, because people didn't care about you
What is the transmogrifier, and what am I?
The transmogrifier is the "multiply by yourself" operation, and you are the number zero.