Circular Logic (Score 3, Funny)

What do we do when the value of 1 starts to float away?
QT/Octave:

>>> sqrt(-1)^((4)^1)
ans = 1
>>> sqrt(-1)^((4)^15)
ans = 1
>>> sqrt(-1)^((4)^16)
ans = 1.0000e+00 - 2.6298e-07i
>>> sqrt(-1)^((4)^17)
ans = 1.0000e+00 - 1.0519e-06i
>>> sqrt(-1)^((4)^21)
ans = 1.0000e+00 - 2.6929e-04i
>>> sqrt(-1)^((4)^22)
ans = 0.9999994 - 0.0010772i
>>> sqrt(-1)^((4)^32)
ans = 0.56253 + 0.82678i
>>> sqrt(-1)^((4)^40)
ans = -0.97448 + 0.22446i

The fractal community will be outraged.