See my other reply down below. That said I agree with the final conclusion that including the i is not a good idea, although I'm a bit on the fence on if the whole thing (I'm convinced by the tau=2pi arguments on the other hand).
I think this really is just a redefinition of sin and cosine (and consequentially the exponential). My feeling is that we now have to deal with different derivative operators for periodic and non-periodic functions and a lot of other weird disconnects, e.g. we don't have a relation between a cycle and the radius (diameter, circumference ...) anymore. I'm not convinced that the (minor IMO) conveniences that gives us is worth it.
I think this really is just a redefinition of sin and cosine (and consequentially the exponential). My feeling is that we now have to deal with different derivative operators for periodic and non-periodic functions and a lot of other weird disconnects, e.g. we don't have a relation between a cycle and the radius (diameter, circumference ...) anymore. I'm not convinced that the (minor IMO) conveniences that gives us is worth it.