User blog comment:Sbiis Saibian/Googology101 - Part II/@comment-10429372-20141024190431/@comment-5982810-20141025010139

It is indeed unbounded. The reason is that we can choose a real part and compute an imaginary part such that  | 10^x | = | x |. That is, we can get values which return results which are the same distance from zero, though possibly at different angles. If we graph the real part as x, and the imaginary part as y, then the function | 10^(x+iy) | = | x + iy | is asymptotic to the function y = 10^x (it isn't identical as there is a small differential at each value of x ). This is why the points on Vels graph appear to be forming an exponential curve. The reason why there are discontinuities is because not every solution results in the same magnitude and the same angle. However as y-->inf, the angle component begins to turn, first slowly, and then exponentially more rapidly. This is why the dots get closer and closer together towards the end (I predicted this behavior before I even got to see Vels graph). It's easy to compute the points on the curve | 10^(x+iy) | = | x + iy |, but finding the points where 10^(x+iy) = x+iy is more tricky as the angle spins around wildly.

From this we can gather a few things. The absolute distance of the fixed points is unbounded, and that if we connect the points with the exponential curve as I suggest, the result will be that it's slope is unbounded as well.