Complex multiplication is very well understood geometrically and algebraically, but I wonder what about the following operators -angles assumed to be randians $[0,2\pi)$:
- Complex multiplication(muladd): $$ x_1 \cdot x_2 = |x_1||x_2|e^{(\arg(x_1) + \arg(x_2))i}$$
- Complex mulmul: $$ x_1 \bigodot x_2 = |x_1||x_2|e^{\arg(x_1) \cdot \arg(x_2)i} $$
- Complex addadd: $$ x_1 \bigoplus x_2 = (|x_1|+|x_2|)e^{(\arg(x_1) + \arg(x_2))i} $$
- Complex addmul: $$ x_1 \bigotimes x_2 = (|x_1|+|x_2|)e^{(\arg(x_1) \cdot \arg(x_2))i} $$
Have these operators been studied? Are there any books or papers on their properties? Do they have names?
I haven't heard of any of those (besides ordinary complex multiplication) being studied. The problem with the ones that multiply arguments is that usually when taking the arg of a complex number, we get an angle out. Angles are cyclical, so that $2\pi$ is treated the same as 0. But multiplying angles doesn't preserve this cyclical nature. $0 \times \frac{\pi}{2}$ is not the same as $2\pi\times\frac{\pi}{2}$. So those operations will be discontinuous. And finally, addadd is pretty much vector addition.