2.86295 + 3.22327 i

[sheldonison]

update: the Wolfram-alpha-contourplots for z^(z-1)-1 from 1+I to 10+10*I  (separate for real and for imaginary parts) give zeros on continuous lines, and an overlay seems to indicate more systematical zeros at the crossings of the lines.                                  

Some small programming in Pari/GP gave the following additional solutions  

Nice work Gottfried.  I could plot sexp for the Kneser solution for a couple of these bases with repelling fixed point b^b=b.... Kneser/fatou.gp uses the primary fixed points, which are different for these bases, but also both repelling.  These tetarion bases are outside the ShellThron region.

Code:

b^b=b; b^L1=L1; b^L2=L2;
b=~2.8630+3.2233*I;  L1=~0.23429+0.72594*I;  L2=~-0.50419-1.0820*I;  
b=~3.7273+5.3180*I;  L1=~0.13364+0.66506*I;  L2=~-0.45567-0.86504*I;
b=~4.4332+7.1938*I;  L1=~0.08774+0.62923*I;  L2=~-0.42709-0.76959*I;

Kneser tetration plot for b=~2.8630+3.2233*I from -3+3i..+8-4i; generated using fatou.gp
   
When iterating exponentials, one tends to quickly get very large numbers or very small numbers.  In this plot, very small numbers have Tet(z)=~0 which is black, and then Tet(z+1)=~1; Red.  And Tet(z+2)=~2.8630+3.2233*I; Orangish.  Since its a repelling fixed point Tet(z+n) will eventually break away from the fixed point.
Continued Kneser tetration plot from +4+3i..+15-4i; Notice how the unstable fixed point takes over.  The white regions are large in magnitude and are seeds for super-exponential growth, but those regions are also unstable since when iterating large complex numbers half the time you go from very large to very small.