Extension of tetration to other branches

[bo198214]

I tried \( [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...] \) (see the obvious pattern), \( b = \sqrt{2} \) as the "infinite branch code". This seems to make it converge onto the constant function with everywhere-value ~9.21189016 + 3.20965748i. It seems any
code will work. E.g. \( [0, 1, 1, 1, 1, 1, 1, 1, ...] \) produces a different constant, ~9.09072986 + 3.37850978i.

I am still not really familiar with those disconnected Riemann surfaces.
Do these constants have a meaning without those limit formulas?
Something path related, infinite paths, fractal paths?