Interesting, but I wonder if you’d get all the same benefits by just measuring angles in units of turns instead of radians. That seems cleaner than the weird dbar differential stuff.
You can always define angles in turns. But the problem is that it conflicts with the definition cos(x) = Re{e^(ix)}. Trig is not so easily separated from the rest of mathematics.
It is possible to completely remove the e^x function from mathematics without losing anything.
It is possible to express everything using a pair of functions, the real function 2^x and the complex function 1^x.
Then the cosinus and the sinus are the real and imaginary parts of 1^x (where x is measured in cycles a.k.a. turns).
The only disadvantage of this approach is that symbolic differentiation and integration are more complicated, by multiplication with a constant.
In my opinion the simplifications that are introduced everywhere else are more important than this disadvantage.
The real and complex function e^x was preferable in the 19th century, when numeric computations were avoided as too difficult and simple problems were solved by symbolic computation done with pen and paper.
Now, when anything difficult is done with a computer, both the function e^x and associated units like the radian and the neper are obsolete.
When programming computations on a computer, using 2^x and 1^x results in simpler and more accurate computations.
Moreover, when doing real physical measurements it is possible to obtain highly accurate values when the units are cycle and octave, but not when they are radian and neper.