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

[Gil]

For my HP50G LambertW program, I was making comparisons with Wolfram Alpha.

In Wolfram Alpha:
LambertW(-100)=3.2+2.5i
LambertW(-100)=3.2+2.7i
LambertW(-1E6)=11.4+2.9i
LambertW(-1E9)=17. 8+2.98i
LambertW(-1E15)=31.1+3.04i
LambertW(-1E30)=64.9+3.09i
LambertW(-1E30000)=69066.4+3.1415i
LambertW(-1E3000000000)=6.9E9 +3.141592653i
...

Two observations, as real x tends to go to very large negative values :
1) we see that the real part (of LambertW0) seems to —> +infinity;
2) we see that the imaginary part (of LambertW0) seems to —> pi.

My questions
If 2) is true, it means that lim IM(W0) =pi≠0, when x —> -infinity.
Then, can we say, considering the above, that lim W0(-infinity) = -infinity, as does write Wolfram Alpha, leaving out the imaginar part?

Or W0(-infinity) = -infinity+pi*i... = -infinity?
Or, more generally:
if lim "A" = C, when x-> B,
& C=±infinity +i×(constant K),
can we say that the limit C=±infinity,
leaving out the imaginary part (the constant K)
(the latter, K, being "peanuts" relatively to the ± infinity of the real part)?

I thank you in advance for your comments.

Gil