Let's be explicit: consider $z=x+iy$ . Then we have $ \frac{z}{\bar z} = \frac{x+iy}{x-iy} = \frac{x^2-y^2+2ixy}{x^2+y^2} = 1 + \frac{-2y^2+2ixy}{x^2+y^2} = -1 + \frac{2x^2+2ixy}{x^2+y^2} ~~, $ the latter two appearing by adding and subtracting in one case $x^2$, in the other $y^2$ from the top of the fraction.
When we say $z\to\infty$, as you mentioned this means that the norm of $z$ must grow without bound, and so $x^2+y^2 \to \infty$. However, there are many, many ways for this quantity to grow to infinity; for instance, take the simple cases of traveling to the right along the $x$-axis (so $y=0$, $x\to\infty$) and traveling up the $y$-axis (so $x=0$, $y \to\infty$) . Clearly both satisfy $x^2+y^2 \to \infty$ .
But notice that $ \lim_{y=0,x\to\infty} 1 + \frac{-2y^2+2ixy}{x^2+y^2} = 1 ~~, $ while $ \lim_{x=0,y\to\infty} -1 + \frac{2x^2+2ixy}{x^2+y^2} = -1 ~~. $ Hence, the limit cannot exist. The axes are just the simplest choices -- taking different paths to infinity can give you different limits.