Complex numbers can be extended using the polar form
$$ Ƶ=re^{i\theta}$$
This allows the complex number to sit on a unit circle, but this is only if theta is a real number. But what if angle theta is imaginery or complex?
What does having an imaginary angle even mean, is there an intuitive way to understand it?
$$ \theta = \alpha + \beta i$$ $$i \theta = i\alpha - \beta $$
$$ e^{i\theta} = e^{-\beta } e^{i\alpha} = e^{-\beta } (\cos\alpha+i\sin\alpha)$$
The point is not on the unit circle anymore.
Assume $r>0$. Instead of looking at $r\exp(\mathrm i\theta)$ replace $r$ by $\log r$ and you get $z=\exp(r+\mathrm i\theta)$.
Also $\exp(a+\mathrm i b) = \exp(a)\exp(\mathrm i b)$. This way you have an identity that simply expresses each complex number by $\exp(z)$ for some complex $z$.
Now in your case: If $\theta$ is complex, let’s say, $\theta = a+b\mathrm i$. Then $\mathrm i\theta= \mathrm i a - b$. So $$ r\exp(\mathrm i\theta) = r\exp(-b)\exp(\mathrm i a) $$
Thus there does not really exist an imaginary angle. $\exp$ is the homomorphism that maps $\mathbb C$ to $\mathbb C^\ast$. Then the angle is basically the imaginary part of the inverse.
