Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I)

[Ivars]

I wondered what results can be obtained if we calculate 2^I , 2^(2^I) , etc.

\( 2^I = e^{I*\ln2} \)
\( 2^{(2^I)} = 2^{e^{I*\ln2}} \)

\( e^{I*\ln2}= cos (I*\ln2)+I*sin(I*\ln2) \)
\( 2^{2^I}= 2^{cos (I*\ln2)+I*sin(I*\ln2)} \)

etc up to infinity of power tower.

Also, another interesting bases:

\( \phi^I, \phi^{\phi^I}... \)

Golden mean may be interesting since \( I/2=sin(I*\ln(\phi)) \)

\( e^I, e^{e^I}, e^{e^{e^I}}, ... \)

\( \Omega^I,\Omega^{\Omega^I}... \) where \( \ln(\Omega)=-\Omega \)

And in general a and z as basis.

I checked with Pari precision 200:

\( \Omega^{\Omega^I,...}=0.68000247... \)

\( \ln2^{(ln2^{(ln2^I)}...} = 0.757558... \)

\( {1/e}^{1/e^{1/e^{...I}}} = 0.567143...= \Omega \)

but \( \Omega^{1/\Omega}= 1/e \) so this iteration where it converges leads to numbers whose self roots is base a - for real numbers - as IF there was no I on top.. Like normal tetration

Would You know of anyplace where such results would be published, e.g graphically on complex plane?

Ivars