I think I know what you're getting at, but let me know if I get it wrong (assuming you're still invested in an explanation!).
I don't know which specific mapping the paper chose. It's possible they chose the mapping in which each rotation gets mapped to the appropriate quaternion with a positive scalar (real) component. In that case, the second quaternion would be $0.079121-0.996865\mathbf{k}$, which is (as advertised) far off from the corresponding quaternion at $3.1$ radians.
The underlying issue is that the quaternions are a double cover of $SO(3)$, and for the mapping from $SO(3)$ to $\mathbb{H}^*$, you have to choose which quaternion goes to each rotation. No matter what choices you make, you will end up with a neighborhood of rotations somewhere that has widely divergent quaternions.
A similar thing happens if you try to define a square root function on the complex plane. Here, again, you have two choices for $\sqrt{z}$ if $w^2 = z$: $w$ and $-w$. If you were to choose whichever one has positive real component (by analogy with what we just did with the quaternions), you end up with a discontinuity at $z = -1$, no matter whether you pick $i$ or $-i$.
Nor can you avoid the problem by making other choices. Imagine traversing the unit circle counterclockwise from $1$, back to $1$. Whatever choice you make for your square root, traverses the same circle in the same direction, but at half the speed. So suppose you choose $\sqrt{1} = 1$ (sensibly). Then you have
$$
\sqrt{i} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i
$$
$$
\sqrt{-1} = i
$$
$$
\sqrt{-i} = -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i
$$
and lastly, $\sqrt{1} = -1$. Oops. We've made $z$ make a full turn around the origin, but its square root has only made a half turn.
Nor can you even fix this by making the square root multi-valued, because now you have to decide which value comes "first." The best you can say is that both values are valid square roots, but you can't make square root a continuous function.
Anyway, we can make the same problem happen in the quaternions by looking at the mapping of rotations of $\theta$ around the $z$-axis, all the way from $\theta = 0$ to $\theta = 2\pi$. Whatever you do, you either have a discontinuity somewhere along the way, or else you end up swapping $1$ for $-1$.