Plotting the function on the real axis reveals that it has three solutions
Therefore the solution set is:
$$ z = \left\{-\frac{2}{\ln2}W\left(\frac{\ln2}{2}\right),2,4\right\} $$
But plotting in the complex plane reveals that there are other solutions far away from the real axis.
The numerical approximation result is:
$$ z≈7.65±11.95i $$
How should we give the expression for the complex solution of the original equation?
The Lambert $W$ function is multivalued, with infinitely many branches. The one you're taking as $W(\ln(2)/2)$ is the "$0$" branch. The complex roots correspond to the other branches. Branches $1$ to $5$ correspond to complex roots approximately
$$ 7.654506496- 11.95367891 \,\mathrm{i}, 10.03684857- 30.81806330 \,\mathrm{i}, 11.31574563- 49.20372800 \,\mathrm{i}, 12.19872547- 67.46928744 \,\mathrm{i}, 12.87413263- 85.68453064 \,\mathrm{i} $$
and $-1$ to $-5$ the complex conjugates of those.