I've stumbled upon an operation (let's call it $\star$) on two complex numbers that performs
$$ (a + bi)\star(c + di) = ac + bdi $$
a couple of times now. Does this operation have a name? And is it an important operation or is it just by chance that I've stumbled upon it?
You see it more commonly described as the componentwise multiplication on the set $\Bbb R\times \Bbb R$. The set $\Bbb R\times \Bbb R$ with componentwise sum and product is usually referred to as the direct product ring of $\Bbb R$ and $\Bbb R$, and in ring theory the notation $\Bbb R\times\Bbb R$ universally refers to it.