Lambert function and Wolfram or "±infinity+i×K=±infinity" ?

[Gil]

As I said,
LambertW (0,-infinity) = (infinity) with Wolfram Alpha.

I think it is impossible, as the (given answer -= infinity) * EXP (given answer =infinity) can't be equal to -infinity

Can Wolfram Alpha be wrong?

Instead, the answer seems rather to be (infinity+i*pi).
Then (infinity+i*pi) × exp(infinity) × cos (pi) =
Infinity × -1= -infinity.

But infinity + i*pi = infinity + number = infinity...

In a way, Wolfram may correct.

Question, what most "reasonable" output should I give for W0(-infinity), W0(±i×infinity), W0(±infinity±i×infinity)?