Two plots of the truth value of sqrt(z^2-1)*sqrt(z^2+1) = sqrt(z^4-1) in the complex plane

From the Maple Conference 2021 Art Gallery

The images concern this potential simplification that a Maple user could make:

While valid for real z this identity does not hold for some complex values of z. The green areas
of the images show where it is true and the red areas where it is false (the image does not
clarify the truth on the boundaries). The images were produced by decomposing the complex
plane according to the branch cuts of the functions in the identity, which may be obtained from
Maples FunctionAdvisor tool. Both images are cylindrical algebraic decompositions of these
polynomials, with the second showing a more efficient decomposition from the algorithm in http://dx.doi.org/10.1016/j.jsc.2015.11.002.