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.