To find the correct figure that represents a $45^\circ$ clockwise rotation of figure (I), we need to visually examine how the patterns in figure (I) would shift when rotated.

1. Understand the Rotation: A $45^\circ$ clockwise rotation means the entire figure will be turned to the right, leading to a diagonal alignment compared to its original layout.
2. Analyze Figure I: Observe the unique corners, edges, and internal patterns in figure (I). These will help identify how the elements are rearranged after rotation.
3. Compare with Options:
   • P: This figure does not correctly represent the $45^\circ$ rotation as the arrangement of the elements does not match the necessary diagonal shift.
   • Q: This figure accurately shows how the components of figure (I) redistributes when rotated. The diagonal alignment of patterns fits perfectly with what is expected from a $45^\circ$ clockwise turn.
   • R and S: Both figures show misalignment in the post-rotation position of elements, implying incorrect rotation outcomes.
4. Conclusion: Based on the analysis, option Q accurately represents a $45^\circ$ clockwise rotation of figure (I).