Let $k\geq 1$ and let $f=\sum_{n\geq 1}a(n)q^{n}$, $a(n)\in\mathbb{R}$, be a normalised cuspidal Hecke eigenform of weight $2k$ for $\Gamma_{0}(N)$ without complex multiplication. So the result of Barnet-Lamb et al. tells us that the numbers $\frac{a(p)}{2p^{k-1/2}}$ are $\mu$-equidistributed in $[-1,1]$, where $\mu$ is the probability measure on the interval $[-1,1]$ defined by $\frac{2}{\pi}\sqrt{1-t^{2}}\,dt$, and $p$ runs through the primes not dividing $N$.

I am looking for any generalisation of Sato-Tate conjecture when Fourier coefficients $a(n)$ are complex numbers, $a(n)\in\mathbb{C}$?