Could you give me an example of two non-isomorphic groups with the same complex character table?
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Once you have one example of nonisomorphic groups $G_1$ and $G_2$ with the same character table, the groups $G_1 \times H$ and $G_2 \times H$ also have the same character table for any finite group $H$ (because irred. characters of $G \times H$ are precisely products of irred. characters of $G$ and irred. characters of $H$). If you take for $H$ a group with order relatively prime to the common order of $G_1$ and $G_2$, then $G_1 \times H$ is nonisomorphic to $G_2 \times H$. So you get infinitely many examples that way.
The example of the dihedral and quaternion groups (of order 8) mentioned by Gerry is the starting point for an infinite sequence of examples: the dihedral and generalized quaternion groups of order $2^n$ for all $n \geq 3$ are nonisomorphic with the same character table.