In a quaternionic plane there are 2 axes and each point corresponds to $q \in \mathbb C^2$. Now, $|z_1|^2 = |z_2|^2$ should define a surface that divides a 3-sphere $|z_1|^2 + |z_2|^2 = 1$ into two pieces, where $z_1, z_2 \in \mathbb C$. What does this surface look like? Describe the two pieces.

We know that a point say $q \in \mathbb C^2$ defines a circle, and in the special case where $q \in S^3$ then it should be a fiber in Hopf fibration.

We also know that the intersection of $|z_1|^2 = |z_2|^2$ and $|z_1|^2 + |z_2|^2 = 1$ consists of 4 unit quaternions, which under stereographic projection also defines the 16 vertices of a 4-cube in $\mathbb R^3$.

It is too difficult for me to see how the surface $|z_1|^2 = |z_2|^2$ cuts an $S^3$ into 2 parts, if anything it should cut it into 4 parts, (referring to the quaternionic plane where $S^3$ is represented as a circle.)

Reference: Zachary Treisman (2009) exercise 2.23. arXiv:0908.1205 [math.HO]