TITLE: How to raise a number to a quaternion power
QUESTION [19 upvotes]: Now, I know that it's (relatively) easy to calculate, say, $r^{a+bi}$ (using the fact that, for  $z_1, z_2\in \mathbb{C}, {z_1}^{z_2}=e^{z_2\ln(z_1)}$ and $\ln(z_1$) can just be found using: $\ln(a+bi)=\ln[\sqrt{a^2+b^2}]+i \cdot \arctan(\frac{b}{a})$  ).
Anyway, how would I go about calculating, say, $i^j, k^i$ etc., or, more generally, $(a_1+b_1i+c_1j+d_1k)^{a_2+b_2i+c_2j+d_2k}$  (I know that exponentiating a complex number (to another non-real complex number) produces a non-unique result, so I assume the same would apply further up the hypercomplex ladder; if that's the case, I'm only concerned with the 'principal' value)?
I obviously don't want a general formula or anything like that; just some intuition and a method by which I could calculate such a thing.
And, finally (because I really like to push my luck), can this method for quaternions be extended to higher number systems (i.e. $\mathbb{O, S},$ etc.) to give an analogous result? 
Thanks

REPLY [2 votes]: I do see a point in defining $x$ to the power of $y$ for general $x$ and $y$. It is the following rationale.
The famous Mandelbrot-Set for computer-graphics has an iteration that can nicely be generalized with meaningful results. 
Originally a Julia-Set is generated by a non-divergence criterion on some complex number $z_0$ with respect to a complex parameter $c$. A series 
$$z_{k+1} = z_k z_k + c$$ 
is calculated as divergent or non divergent for $z_0$ given. Whenever $c$ is replaced by the identity-mapping on the Eulerian plane, i.e. 
$$c(z_0) = z_0$$, 
matters simplify and the famous Mandelbrot-Thing appears.
The complex multiplication has a useful square mapping. Whenever a higher exponent than $2$, e.g. $3$, $4$, $5$, or, what you want, is applied, we get a meaningful object of studying a general Mandelbrot-Thing by calculating the divergence of
$$z_{k+1} = z_k \cdot z_k \cdot z_k \cdot\dots\cdot z_k + z_0$$
for $z_0$ around $0$ complex.
This meaningful object has got a non-trivial scale-appearance and an astonishing way, how symmetries resemble this natural exponent, used. This proposed natural exponent increased to great numbers seems somewhat to create an increasingly circle-like fractal in the complex plane, with inner and outer circular limit and with a narrow meander of a fractal curve in between.
The quarternions are the last thing useful for studying this fractal-jazz. Some saying from Euler, I remember cited, however, says, the easiest way to a real problem would make use of complex models. The Zeta-Function discussions for a famous Riemannian millenium-problem might benefit from proper terms for some way to circumvent all the particularities of pure complex models by actually defining everything it takes to work with $x$ to the power of $y$ for general $x$ and $y$.
I will comment to the first answer, if I have 50 reputation. For the time being this text must be part of the answer to the original question about quarternions. So, in brief, the way how to raise a number to a quarternion power, is in what it shall mean to everyone.