Vector operation: add angles, multiply absolue values. How is it called? For two complex values, multiplication amounts to adding their polar-representation angles and multiplying their absolute values.
For $\mathbb{R}^n$ I could define the same rule for two vectors $\vec{x}$ and $\vec{y}$ as $\vec{z} := \vec{x}\curvearrowleft\vec{y}$ such that
\begin{align}
     \operatorname{polarangles}(\vec{z})
  &=   \operatorname{polarangles}(\vec{x})
     + \operatorname{polarangles}(\vec{y})\\
     |\vec{z}|
  &= |\vec{x}| \cdot |\vec{y}|
\end{align}
where $\operatorname{polarangles}(\dots)$ is a tuple of $n-1$ polar coordinate angles.
Since addition and multiplication are commutative and associative, it looks to me as if $\curvearrowleft$ should be too, with the unit vector in $x$ direction ($z$ direction, hmm?) being the neutral element of the operation.
On the other hand the operation contains a rotation which, afaik, is not associative for $n\geq3$. Is it or is it not associative?
Does this operation have a name, is there some theory about it? Or can I have an argument or example showing that it is actually not well defined?
There is a similar question, but the answers all focus on different points. Most suggest, though indirectly, that the operation is useless, but without giving arguments (apart from the distinguished direction argument in the top voted answer).
 A: This attactive idea is unfortunately doomed by the Frobenius theorem and even more by Hurwitz's theorem. The latter says, slightly paraphrased, that if you have a binary operation $*$ on $\mathbb R^n$ that satisfies:


*

*$\|x*y\| = \|x\|\cdot\|y\|$,

*The distributive laws $x*(y+z)=x*y+x*z$ and $(x+y)*z = x*z+y*z$, and

*$(r,0,\ldots,0)*x = x*(r,0,\ldots,0) = r\cdot x$ for every $r\in\mathbb R$


then the $n$ is either $1$, $2$, $4$, or $8$, and the whole thing is isomorphic to either the reals, the complex numbers, the quaternions, or the octonions.
In particular, no matter what you do with the angles, you can't get a solution for $\mathbb R^3$ that distributes over vector addition.

What goes wrong in your concrete attempt is that the addition of polar angles is not really well defined. The addition means that you can't restrict the angles to a single interval, so there will be more than one set of angles that represent the same direction, and your componentwise addition does not respect this equivalence.
For example, in $\mathbb R^3$, say we're using longitude and latitude as our polar angles. Then $90^\circ\, E,\; 45^\circ\, N$ is the same place as $90^\circ\, W,\; 135^\circ\, N$ -- but if we add each of these to itself then we end up on the north pole in the first case and on the south pole in the second!
We get a similar problem if we use inclination/azimuth coordinates instead.
Note also that adding something to either latitude or azimuth does not correspond to a rotation of the unit sphere.
