Question: Are the real and imaginary parts of the complex number $(-1/2)^{(-1/2)^{(-1/2)}}$ irrational? Transcendental ?
Background and motivation: I became curious about numbers writable as $$\Bigg |(-1/n)^{(-1/n)^{(-1/n)}} \Bigg|$$ This was motivated by me trying to write $e^{\pi\sqrt{2}}.$ Now Gelfond-Schneider tells us that $e^{\pi\sqrt{2}}$ is irrational and transcendental. On the other hand I know that $(-1/n)^{(-1/n)^{(-1/n)}}$ is a complex number in particular there exist real numbers $z_0$ and $z_1$ such that $$(-1/n)^{(-1/n)^{(-1/n)}}=z_0+iz_1 $$ For example if $n$ is equal to $2$: $$(-1/2)^{(-1/2)^{(-1/2)}}= \color{blue}{47.339658423\ldots}+i\color{blue}{70.6208560994\ldots}$$ I do not believe Gelfond-Schneider shows $47.339658423\ldots$ and $70.6208560994\ldots$ to be irrational or transcendental.