While seeking all solutions of $ Z ^ 2 = 2 ^ Z $ we have three real roots of $Z : z_1=2, z_2=4, $ and a third real root given in terms of LambertW function:
$ z_3=-\dfrac{2 W\left(\frac{\log (2)}{2}\right)}{\log (2)}\approx -0.7666646960 $
The situation is quite unsymmetrical ( or so it appears to me), so may I ask to
find out the fourth complex constant root $z_4$ such that
$$ z_3^{z_{4}} = z_4^{z_{3}} $$
is satisfied.
Reference : A comment by Claude Leibovici to my query regarding above three real roots.
EDIT1:
If $ f(z,a) = z^a - a ^z,$
when $ a=2, $ roots are $ z =(2,4, z_3 ) $
when $ a=4, $ roots are $ z= (2,4, ~ -0.26028 \pm .869545 i) $
EDIT2:
when $ a=e, $ one root only, it is $ z= e $, don't even know if it is repeated!
And why this hinging around $z=e?$
When real constant in the given equation changes real roots do not change but complex parts are changing.
The fourth root is actually written in "closed form" just the same as the third root. Note that $W(x)$ has infinitely many solutions if solutions are allowed to be complex.
This is much like $\ln(x)$ having infinitely many solutions. When we allow the logarithm to do such, we call it the complex logarithm and it has such properties because of $e^{\theta i}=\cos(\theta)+i\sin(\theta)$ allowing $\theta$ to have the property $\theta=\theta\pm2\pi n,n=0,1,2,3,\cdots$