In other words, $z^n \equiv \bar{z}\pmod{n}$ or $z^{n+1} \equiv z\bar{z}\pmod{n}$. That splits into two conditions: $$\begin{cases} \Im z^{n+1}\equiv 0\pmod{n}, \\ (c^2+1)^{n}\equiv c^2+1\pmod{n}. \end{cases} $$
Numbers $n$ satisfying the first condition Grau et al. (2014) called Gaussian Fermat pseudoprimes to base $z$, while numbers satisfying the second condition (under the additional constraint $\gcd(c^2+1,n)=1$) are Fermat pseudoprimes to base $c^2+1$. There are many composite examples satisfying either of the two conditions, it is hard to find those that satisfy both.
For example, Grau et al. showed that there no composite $n=4k+3$ below $10^{18}$ that are both Gaussian Fermat pseudoprimes to base $1+2I$ and Fermat pseudoprimes to base $2$.
Alternatively, it can be seen that $z^n$ can be expressed in terms on Lucas sequences: $$z^n = \frac{1}{2}V_n(P,Q) + I U_n(P,Q)$$ for $(P,Q)=(2c,c^2+1)$, and hence $n$ satisfying the two conditions is a Frobenius pseudoprime.