Pentation's definitional ambiguity

[Base-Acid Tetration]

for every k, there exists a branch such that \( {}^{b}a = c + \frac{2 \pi i k}{\ln(a)} \) is this the same as the indexing scheme above?

Then I would think the number of branches is still countable, though i doubt it because a[4]b 's cardinality is that of the continuum.