1^x is multi-valued, and like with all other multi-valued complex functions it is possible to select a branch of the function that is a proper function.
Defined correctly, the value of 1^x for any rational x is the corresponding smallest root of unity, and for irrational arguments it is defined by continuity.
Therefore 1^x rotates on the unit circle for increasing x.
The standard complex exponential and complex logarithm functions are defined exactly in the same way, because they are also multi-valued, so the same kind of equalities like yours would also be true for them.
With the correct definition, 1^x = 1 only for integer x, not for any x. As I have written above, the useful 1^x is defined only for real arguments, not for complex arguments. Only its values are complex numbers of unit modulus.