Can every possible 'imaginary number' be expressed in terms of $i$? The domain of the function $f(x)=\sqrt{x}$ can be extended to all real numbers by introducing a new number, $i=\sqrt{-1}$.
Can this be done for any function, say $\arcsin{x}$, or $\log{x}$?
What is $\arcsin{2}$? Or $\log{-3}$?

Multiplying with $i$ rotates 90 degrees in the complex plane. One way to be able to reduce that to smaller steps is:
You can use a number really close to positive 1 (just a tiny little bit imaginary). For example:
$$z = \exp\left[\frac{2 \pi \cdot i}{k}\right], \, \text{ $k$ very large integer}$$
Now $z^k$ will be $1$. So you can reach all points on the unit circle with an angular resolution of $k$ steps per rotation. Now what is needed is to get off the unit circle. We can do this with a real radius $$r\in \mathbb R : r = 1+\frac{f}{100}$$ where f is the step in growth percentage. Now we can reach any complex number on a polar grid with the power functions and integer exponents:
$$r^{e_1}\cdot z^{e_2}, \, e_1,e_2 \in \mathbb{Z}$$
