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Here is one way to find a solution, bypassing the complex nature of the problem:

sage: var('z')             
z
sage: f = cos(cos(cos(cos(z)))) - sin(sin(sin(sin(z))))
sage: var('a,b', domain=RR)
(a, b)
sage: M = f(z=a+b*i).real()^2+f(z=a+b*i).imag()^2
sage: minimize(M,[0,0],algorithm='powell',disp=0)
(0.756887137348, 0.61015499692)
sage: M(a=_[0],b=_[1])                           
6.39826537371e-26

You can try various starting points and other algorithms (ncg,bfgs) - they give different results.

In Maple things are much simpler:

fsolve(cos(cos(cos(cos(x)))) = sin(sin(sin(sin(x)))),x,complex);

-0.7401150735 - 1.364789582 I

so it is possible...