Many of you may recall "An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?", an old question of mine, and just recently, I saw this new question that poses a simple extension to tetration at non-integer values:

$$a\uparrow\uparrow b=\begin{cases}a^b,&b\in[0,1]\\a^{a\uparrow\uparrow(b-1)},&b\in(1,+\infty)\\\log_a(a\uparrow\uparrow(b+1)),&b\in(-\infty,0)\end{cases}$$

Combining this with my old question, I made the following graph of $i\uparrow\uparrow x$ for $x\in(-2,9)$, using $z=re^{i\theta},\theta\in[0,2\pi)$:

(x-axis is the real axis and y-axis is the imaginary axis)

Is there anything special to be said about this?

About how each look always connects to the previous 'branch' and then steps off the branch until it hits the next one. Can we prove this is indeed the case?

Prove or disprove that the tangent line at each interception is equivalent for the original branch and the branch coming out that heads towards the center.

It also appears the branches connect perpendicularly.

Are these shapes similar to one another?

The pattern is rather intriguing, don't you think?

It appears $(-1)\uparrow\uparrow x$ is likewise interesting to look at:

It starts off how one might expect it to start off:

It makes a loop:

Then another loop:

And then it blows up into a circular shape that reaches about 35 units away from the origin:

Closer image:

And then it goes on past $10^{30}$:

Closer image:

Medium zoom:

Another loop:

Any explanation for why this is so much more 'chaotic' than $i\uparrow\uparrow x$? Perhaps we can define chaotic or not as follows:

\begin{align}\lim_{x\to\infty}a\uparrow\uparrow x\ \text{converges}\implies\text{stable/non-chaotic}\end{align}

\begin{align}\lim_{x\to\infty}a\uparrow\uparrow x\ \text{diverges}\implies\text{unstable/chaotic}\end{align}

Once again we see apparently perpendicular connections, though trivially all at $(-1,0)$.

There are new patterns though. We get almost cardioids, but not quite. We saw two interesting loop looking shapes as well. Any idea what these are? It appears these loops get really long and form the quasi-cardioids.

Is it the case that these loops always connect back to $(-1,0)$ from the same direction from which they came?

And is there a 4-turned patter? The first line connecting to $(-1,0)$ came from above, the second line connecting to $(-1,0)$ came from the right, the third came from below, the fourth from the left, and if we keep graphing more of these, the fifth comes from above once again.

Can we do an analysis to these different shapes? My graphing calculator isn't the best, and I have to do these one by one... Particularly, what can we say about $z\uparrow\uparrow x$ for $|z|=1$ and $x\in\mathbb R$?

After quite a few graphs, I've come to the following conclusion that:

When $|z|=1,\operatorname{arg}(z)=\theta\in(-,\pi]$, then $\lim\limits_{x\to\infty}z\uparrow\uparrow x$ tends to exist for $\theta<\theta_0$ and diverges for $\theta>\theta_0$. What is this $\theta_0$? And is my conclusion correct?

It also appears that it may be far from trivial proving the lines in $i\uparrow\uparrow x$ actually connect. Though they are close, I have found that $\sqrt i\uparrow\uparrow x$ clearly does not connect:

Here is the general graph. It appears as though $\theta_0=\pi/2$.