What is this flower? The small, double purple blooms offer a clue
The photograph shows a low, trailing potted plant with many small, double purple flowers. Its growth, leaves and flower shape together make it fairly recognisable.

The flower is a double petunia. It has the typical petunia appearance: funnel-shaped flowers, slightly sticky narrow leaves and trailing growth, here in a small-flowered double form common in summer containers and hanging baskets.
Its dense, small-flowered appearance also resembles calibrachoa, but the double flowers and overall impression fit petunia better in this photograph. I would therefore call it a petunia, specifically a trailing double petunia.
What to ask for in a shop
Ask for a purple trailing double petunia or a small-flowered double petunia. The photograph cannot reliably establish the cultivar, but the plant group is clear enough to help you find something very similar.
Sources and images 1
- Original reader contribution ↗Reader question