Funny how the so-called eldritch terror is another face of what is widely considered one of the most beautiful equations in mathematics, Euler's identity that unites five fundamental constants.
e^(i*pi)+1 = 0
..which is a result of the more general formula.
e^(i*x) = cos(x) + i*sin(x)
Pi is hiding there in the sin and cos functions implicitly, because the unit radian is defined by 2*pi. In comparison, the version you mentioned that takes x in "turns".
-1^(2x) = cost(x) + i*sint(x)
It got rid of pi and e, which already seems a win for simplicity. i is still there for the imaginary component, or y in the complex plane. So the need for pi was removed thanks to the "turn", defined by 1 as the whole circle or cycle.
Multiplying -1 to itself every half turn makes it an alternating series of 1 and -1.. Weird, but it is visually clear to understand, without involving e. Though I still don't see where e went. Oh, this comment explains:
> If we rearrange the products in the exponent we get
2πix πi2x ( πi ) 2x
e -> e -> (e )
> Where e^(πi) is -1. That shows there is something to the turns units; we can express the analog of the Euler identity using exponentiation using a base and factor which are integers.
Yeah I get it now, a "turn" acts like a dimensionless unit to the circle/cycle.