4. Let $z$ be a complex number. If the equation $\left|z^{2}\right|-\left|z^{2}-9\right|=7$ represents a conic section, then the eccentricity $e=$

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Let $z=x+y \mathrm{i}, x, y \in \mathbf{R}$, then $x^{2}+y^{2}-\sqrt{(x-3)^{2}+y^{2}} \sqrt{(x+3)^{2}+y^{2}}=7$ $\Rightarrow x^{2}+y^{2}-7=\sqrt{\left(x^{2}+y^{2}+9\right)^{2}-36 x^{2}} \Rightarrow 36 x^{2}=\left(x^{2}+y^{2}+9\right)^{2}-\left(x^{2}+y^{2}-7\right)^{2}$ $=32\left(x^{2}+y^{2}+1\right) \Rightarrow x^{2}-8 y^{2}=8$, so the eccentricity of the hyperbola $e=\frac{3 \sqrt{2}}{4}$.