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

[Gil]

With Wolfram Alpha
LambertW (k=any value, z=(any real value ±i*infinity) = infinity.

Now it seems that lim LambertW, k free, for z=a+ib & with a=±infinity, b=±infinity, gives equally infinity.

Could anyone, if possible a "math guy", confirm it?

My questions are related of course to HP programs, as I want to implement an output in my Lambert HP50G function for any real or complex "numbers, including the ± infinity input cases, the outputs of which I wish to treat as follows :

Wk=0±1...(oo+ib): 'oo'
Wk=0±1...(a+ioo): 'oo'
Wk=0±1...(±oo±ioo): 'oo'

with oo being the infinity sign.

Thanks again for your insight.