In this paper we introduce an $[m,C]$-isometric operator $T$ on a complex Hilbert space $\mathcal H$ and study its spectral properties. We show that if $T$ is an $[m,C]$-isometric operator and $N$ is an $n$-nilpotent operator, respectively, then $T+N$ is an $[m+2n-2,C]$-isometric operator. Finally we give a short proof of Duggal's result for tensor product of $m$-isometries and give a similar result for $[m,C]$-isometric operators.