Example of a complex transcendental number? Researching transcendental numbers I have only come across ones with a transcendental real part. I can't think of any which are pure imaginary or are not based on a real transendental number, t, of the form $t + ni$. Any ideas?
Edit: After some excellent responses, I now have another thought see Complex transcendentals not known in component form?
 A: Real transcendental number: $\pi$, $e$, $\ldots$ 
Imaginary transcendental number: $i\pi$, $ie$, $ie^\pi$ , $i\pi^e (\text{suspected to be true but not yet proved})$ $\ldots$ 
Complex number with transcendental real and imaginary parts: $\pi+i\pi$, $e+i\pi$, $\pi+ie$, $e+ie$, $\ldots$ 
Moreover, a purely imaginary transcendental number can be $\ln(-1)$, following from the equation $e^{i\pi}=-1$ and in which case, we have to extend the domain of definition and hence the range of the natural logarithm.
A: We have $a + bi$ is algebraic iff $a$ and $b$ are algebraic.
Therefore, if $a + bi$ is transcendental then at least one of $a$ or $b$ is transcendental.
So, all complex transcendental numbers are "based" on real transcendental numbers.
A: I've always been intrigued by
$$i^i=e^{-π/2}$$
(at least its principal value) and you will note that
$$i^{i^i}=e^{(iπ/2) e^{-π/2}}$$
does not fall into the facile category of $a+ib$ where $a$ and $b$ are well-known transcendental numbers.
A: $i^{i+n}$ is transcendental for integer $n$.
