I'll defend i^4x since I like it better.

(cost x, sint x) is a point on the unit circle x turns counterclockwise from (1,0).

cost x + i sint x is a point in the complex plane x turns counterclockwise from 1.

Now look at integer powers of i, a point in the complex plane 1/4 turn from 1:

i^0 = 1 (0 turns from 1)

i^1 = i (1/4 turn from 1)

i^2 = -1 (2/4 turn from 1)

i^3 = -i (3/4 turn from 1)

and we define complex exponentiation such that, for all real x,

i^x = cost (x/4) + i sint (x/4) (x/4 turn from 1)