Why can't we have a third number line for dividing by zero?

This is good, creative thinking about number systems. But here's the hitch: you have to follow through with the consequences of your definitions. If $z = 1/0$ as you suggest, then it must be that $z \cdot 0 = 1$.

Why is this a problem? Well, you are probably already familiar with the idea that $a \cdot 0 = 0$ for any number $a$. Incidentally, this is a consequence of the fact that $0$ is the additive identity and that your operations of multiplication and addition satisfying a distributive law. To wit,
$$ a \cdot 0 = a \cdot 0 + a \cdot 0 - a \cdot 0 = a \cdot (0 + 0) - a \cdot 0 = a \cdot 0 - a \cdot 0 = 0 $$ With $a = z$, your special reciprocal of $0$, you now have that $z \cdot 0 = 1$ and $z \cdot 0 = 0$. Thus, $1 = 0$.

Once you establish that $1 = 0$ in your number system, then any number is equal to any other number! Consider two numbers $b$ and $c$, and calculate: $$ b = b + 0 = b + 0 \cdot (c-b) = b + 1 \cdot (c-b) = c. $$

All your numbers collapse into a singularity. Oh noes!

Interesting question! Although I’m not sure Rafael Bombelli created the imaginary axis since his life largely predates the widespread use of Cartesian coordinate systems.

Before postulating a division by zero axis, it may help to revisit why x/0 has consensus as an undefined concept. It’s not just a void of mathematics, its use leads to contradictions within our other established axioms:

It follows from the properties of the number system we are using (that is, integers, rationals, reals, etc.), if b ≠ 0 then the equation a / b = c is equivalent to a = b × c. Assuming that a / 0 is a number c, then it must be that a = 0 × c = 0. However, the single number c would then have to be determined by the equation 0 = 0 × c, but every number satisfies this equation, so we cannot assign a numerical value to 0 / 0

Lots more contradictions in the wiki, over many different areas of math https://en.wikipedia.org/wiki/Division_by_zero

Which then comes around to, what are we hoping to postulate a new axis or number line to achieve? Exploratory math, sure go ahead, but I’d like to tickle your brain more on why you’d even want to do this.