Why don't we define $\omega$ such that $\omega * 0 = 1$? We define $i$ to be $i^2 = -1$. Can we also define $\omega$ such that if it is multiplied by $0$, we get $1$?  Could we perform operations on it just like we do on imaginary numbers? If so, does $\omega \in \mathbb{C}$ or not?
I read Why can one multiply by zero and Max's answer suggests that the answer to my last question is no.
I am not sure about the tags I have given to the question. Please edit them if needed.
 A: This might not be an answer but it is too long to be posted as a comment.
So if I read you right you want to add an element $\omega$ to $\mathbb{C}$ so that $\omega \times 0 = 1$. You can do so! Take any element not in $\mathbb{C}$, denote it $\omega$ and fix $\mathbb{C}' = \mathbb{C} \cup \{\omega\}$. As you did, denote by $*$ your new "multiplication", $x*y$ is defined as the complex product if $x$ and $y$ are both complex ; as $0$ if $\{x,y\} = \{0,\omega\}$ ; and whatever you want in the remaining case.
Now $\mathbb{C}'$ is a set endowed of a "multiplication", which turns to merely be an application from $\mathbb{C}' \times \mathbb{C}'$ to $\mathbb{C}'$. But what axiom does it satifsfy? Clearly nothing interesting. As mentioned in the first comment the assiociativity is not granted.
The set of complex numbers is (at least, I am not an algebrist) a field. The set $\mathbb{C}'$ endowed of $*$ is not a group so you cannot perform operations as you do in $\mathbb{C}$.
Moral of the story : the operation you want to add can be supported but destroys the algebaic structure which allows you to perform nice computations.
