Let A be a unital not-necessarily commutative algebra, defined over R or C. Take some...

If $\alpha$ is an algebra automorphism, even one that is not assumed to take identity to identity, the multiplicative property of the map necessarily makes $\alpha (1)$ the identity of the image (which is equal to $A$) and so $\alpha (1)=1$.
So there are no algebra automorphisms that are strictly non-unital in the sense that they move the multiplicative identity. $1-\alpha (1)=0$ for every algebra automorphism, and it is always non-invertible.