How to get latitudinal tori using the inverse Hopf map? We define the Hopf map as a function from $S^3$ into $S^2$, $f(z_1, z_2) = \frac{z_2}{z_1}$, where $S^3=\{(z_1, z_2) \in \mathbb C^2 | |z_1|^2 + |z_2|^2 = 1\}$ and $S^2$ is the Reimann sphere $z=\frac{z_2}{z_1}$.
Now, we want to find the inverse map $f^{-1}(z)$, (in order to find the latitudinal tori later.)
$f: (z_1, z_2) \to z$
$f^{-1}: z \to (z_1, z_2)$
If we use $z_1 = \frac{z_2}{z}$ and $z_2 = zz_1$, then we should be able to replace $z_1$ and $z_2$ to find the inverse map, and so we get $f^{-1}(z) = \frac{zz_1}{\frac{z_2}{z}} \Rightarrow f^{-1}(z) = z$, which helps little.
It is understandable that a function with two variables as input and one variable as output is so unlikely to be invertible. However, the input variables $z_1$ and $z_2$ are related since $|z_1|^2 + |z_2|^2 = 1$. Moreover, we will need an inverse Hopf map for showing that the tori $\sigma(f^{-1}(|z|=r))$ are tori of revolution. So there should be an algebraic way for getting an $S^1 \times S^1$ surface using the inverse Hopf map.
We know that in the Reimann sphere, $|z|=r$ defines a circle with the center point at the origin and radius $r$. On the other hand, by sending $z$ back to $\mathbb R^4$ using the inverse stereographic projection and applying $S^1$ action we can get a circle worth of unit quaternions in $S^3$. So, we have our $S^1 \times S^1$ in this way here, but how do they relate in an algebraic way to get a torus of revolution such that $(x_1^2 + x_2^2 + x_3^2 + R^2 - r^2)^2 = 4R^2(x_1^2 + x_2^2)$, where $S^3=\{(x_1, x_2, x_3, x_4) \in \mathbb R^4 | x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1\}$, $r=x_3^2 + x_4^2$ which is the smaller radius and $R=(x_1^2 + x_2^2)^2 \pm \sqrt {r^2 - x_3^2}$ which is the bigger radius.
Found $r$ starting from $z = \frac{z_2}{z_1}$ and using $|z| = r$, next, found $R$ starting from $(|z_1| + x_3^2 + R^2 - r^2)^2 = 4R^2|z_1|$ and using $r = |z_2|$.
Reference: 


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*Treisman, Z., a young person's guide to the Hopf fibration, exercise 4.7.,   (2009), arXiv:0908.1205 [math.HO]

*The Hopf fibration visual model using Geogebra 3D calculator



 A: 
It is understandable that a function with two variables as input and one variable as output is so unlikely to be invertible. However, the input variables z1 and z2 are related since $|z_1|^2+|z_2|^2=1$.

Well, yes, but they're not that related. You can't solve for $z_2$ in terms of $z_1$, only its magnitude not its phase. Consider the real variables involved; the domain $S^3$ is three-dimensional and the range $S^2$ is two-dimensional and the Hopf map is continuous so you can expect (and by "invariance of domain" in topology) the function cannot be invertible.

Moreover, we will need an inverse Hopf map for showing that the tori are tori of revolution.

That's not correct. The notation $f^{-1}$ here is being used for inverse image (rather than "image under inverse function"). The fiber of a point $z_1/z_2$ is the set $\{(e^{i\theta}z_1,e^{i\theta}z_2)\}$ which is indeed the orbit of $(z_1,z_2)$ under the action of $S^1$; the fibers are all circles. However, if you take a circle in $S^2$ and take the inverse image, you get a circle of circles, which is a torus. It turns out the fibers are "diagonal" Hopf circles in the torus, not your usual poloidal or toroidal circles.
Such a circle in the Riemann sphere $S^2$ we can say is $\{e^{i\phi}z\}$ so the inverse image will be the torus orbit $\{(uz_1,vz_2)\mid u,v\in S^1\}$ (this is an orbit of $S^1\times S^1$ acting on $S^3$ within $\mathbb{C}^2$). I'll let you verify under stereographic projection this is a torus of revolution.
