AFAICT, there is an interesting repetition going down the real axis for infinity. From:
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// hyperbolic tan forest
z = (std::sin(z) * std::tanh(z));
     
if (z.real() * z.real() + 
    z.imag() * z.imag() > 23456.0)
{
    // bail
    return;
}
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The higher the escape, the more repetitions occur.