Recall from Opening the Door
that adding a complex number to another number is the same as a translation.
Recall from Into the Wilderness
that multiplying a complex number by another is the same as a rotation and an enlargement.
Since $u+v=v+u$ and $uv=vu$, we can decide which complex number we want to 'transform'. For example, for $u+v$, we could start at $u$ and apply a translation corresponding to $v$, or we could start at $v$ and apply a translation corresponding to $u$.