Could we "invent" a number $h$ such that $h = {{1}\over{0}}$, similarly to the way we "invented" $i=\sqrt{-1}$? $\sqrt{-1}$ was completely undefined in the world before complex numbers. So we came up with $i$. 
$1\over0$ is completely undefined in today's world; is there a reason we haven't come up with a new unit to define it? Is it even possible, or would it create logical inconsistencies?
 What would be the effect on modern math if we did so?

It isn't mathematically consistent with our concepts of multiplication. However, in a similar vein of reasoning, mathematicians sometimes employ the extended real numbers, which are basically the real numbers with two new elements, positive and negative infinity. Although it still isn't 1/0, defining these two numbers has a lot of use in measure theory.
