Most elegant quaternionic function that is odd and multiplicative

Question

I have to set an example and I'd like to have a nice function from the quaternions to the reals such that $$f\left(x*y\right)=f\left(x\right)f\left(y\right)$$ and $$f\left(-x\right)=-f\left(x\right)$$ clearly $*$ is the quaternionic product and juxtaposition is the real product. Essencially I want an homomorphism such that $f(-1)=-1$. Do you have any idea in mind?

Answer 1

The only odd multiplicative map from the quaternion to the reals is the zero map.

Indeed, we have (taking into account that multiplication in $\mathbb{R}$ is commutative):

$f(ij)=f(i)f(j)=f(j)f(i)=f(ji)=f(-ij)=-f(ij)$, so $f(ij)=0$.

But now for any quaternion $q$, we have $f(q)=f(ij (-ijq))=f(ij)f(-ijq)=0$.

In particular,you cannot have $f(-1)=-1$...