exponentiation - What do we actually mean by raising some number to an imaginary power?

It seems quite intuitive when we say that some number $a$ is raised to a power $b$ where $a \in \mathbb{C}$ and $b \in \mathbb{Z}$ and can be expressed as

$$a^b = a \times a \times a ... \text{($b$ times)}$$
Extending the argument such that $b \in \mathbb{R}$ then if $b$ is rational, it can be expressed in the form $\dfrac{p}{q}$ such that $p,q \in \mathbb{Z}$ and $q \ne 0$ and $a^b$ is defined as
$$a^{\frac{p}{q}} = \sqrt[q]{a^p}$$
If $b$ is irrational then $a^b$ is a transcendental number as stated by Gelfond- Schneider theorem ($a$ and $b$ are algebraic numbers). Agreed.

Now, here is the problem: What happens when $b$ is an imaginary number? What is an intuitive idea behind saying $i\theta$ times in the expression (I may be wrong in saying that)

$$e^{i\theta} = e\times e\times e...\text{($i\theta$ times)} = \cos \theta + i\sin \theta$$
Yes, thats the Euler's theorem.