How to plot the Worm domain?

While reading Krantz' "Harmonic and Complex Analysis in several variables" I stumbled upon the Worm domain. It is a counterexample to the long-believed statement that a smoothly bounded pseudoconvex domain will have a Stein neighborhood basis, which was found by Diederich and Fornæss.

I will write the definition of the worm as well of an alternative version. I'd like to visualize them and manipulate parameters.

The definition of the Worm domain might be a little bit longer since I will also quote a small but relevant part of the book. The other version is more comprehensive.

Definition (Worm). Let $\mathcal W$ denote the domain $$\mathcal W=\left\{(z_1,z_2)\in\Bbb C^2 : \left\lvert z_1-e^{i\log\lvert z_2\rvert} \right\rvert^2<1-\eta(\log \lvert z_2\rvert^2)\right\},$$ where (i) $\eta\colon\mathbb R\to\mathbb R,\eta\ge0, \eta$ is even, $\eta$ is convex; (ii) $\eta^{-1}(0)=I_\mu=[-\mu,\mu]$; (iii) there exists a number $a>0$ such that $\eta(x)>1$ if $\lvert x \rvert>a$; (iv) $\eta'(x)\neq0$ if $\eta(x)=1$.

"Notice that the slices of $\mathcal W$ for $z_2$ fixed are discs centered on the unit circle; the centers of these circles wind centered on the unit circle; the centers of these circles wind $\mu/\pi$ times about that circle as $\lvert z_2\rvert$ traverses the range of values for which $\eta(\log\lvert z_2\rvert^2)<1$. It is worth commenting here on the parameter $\mu$ in the definition of $\mathcal W$. The number $\mu$ in some contexts is selected to be greater than $\pi/2$. The number $\nu=\pi/2\mu$ is half the reciprocal of the number of times that the centers of the circles that make up the worm traverse their circular path."

An alternative version of the Worm is a simplification of the definition above. We can take $\eta$ to be $1$ minus the characteristic function of the interval $[-\mu,\mu]$, which has the effect of truncating the two caps and destroying in part the smoothness of the boundary.

Definition (Non-smooth version of the worm).

$$\mathcal W'=\left\{(z_1,z_2)\in\Bbb C^2 : \left\lvert z_1-e^{i\log\lvert z_2\rvert} \right\rvert^2<1,\left\lvert\log \lvert z_2\rvert^2\right\rvert<\mu\right\}$$

Conclusion: In order to get a meaningful plot, I certainly want to fix $\mu$ and $\eta$. I'm not sure what to expect from the plot of the Worm so I should also be able to fix $z_1$ or $z_2$. Maybe it is also possible to plot both $z_1$ and $z_2$ at the same time, using coloring and 3-dimensional space.

The only image of a Worm I was able to find is here one page 2. As I said I'd like to interact with the plot and see exactly how it changes when we choose different parameters.