Suppose we have iid $X_1,\ldots, X_n\sim N(\mu,\sigma^2)$ where $\mu$ is unknown.
Let $X_{(1)}, X_{(2)},\ldots,X_{(n)}$be the order statistics.
Is the order statistics $\textbf{complete sufficient}$ when
$A)$ $\sigma$ is known ?
$B)$ $\sigma$ is unknown ?
Since your question is phrased like a textbook exercise and you haven't said anything about what particular difficulties you've had with it, I will keep this short.
Note that $\operatorname{E}(X_{(n)} - X_{(1)}) = a\sigma,$ for some $a>0,$ and although finding the value of $a$ may be a lot of work, (the following is crucial) you do not need to know $\mu$ or $\sigma$ to find $a.$ In other words, $a$ is "known". Note that $X_{(n)} - X_{(1)} - a\sigma$ is therefore a statistic if $\sigma$ is known, but not if $\sigma$ is not known, and its expected value is $0.$
That should get you started on this.
As for sufficiency, the complete set of order statistics from i.i.d. observations is always sufficient.