There are several e-commerce platforms that offer a high level of customization for UX designers. Here are a few options worth considering:

1. WooCommerce (WordPress): WooCommerce is a popular e-commerce plugin for WordPress. It offers a wide range of customization options through themes and plugins. You can easily modify the layout of checkout pages, product overviews, and individual product pages using its extensive theme customization features. With thousands of free and premium themes available, you can find one that suits your needs or create a custom theme from scratch. WooCommerce also provides various extensions to enhance functionality, such as advanced product filters or product comparison tools.

Reference: WooCommerce - https://woocommerce.com/

2. Shopify: Shopify is a leading e-commerce platform known for its user-friendly interface and extensive customization options. It offers a wide range of customizable themes and templates, allowing you to modify the layout of various pages. Additionally, Shopify's theme editor provides a drag-and-drop interface, making it easy to customize the checkout process, product pages, and more. You can also use Shopify's Liquid programming language to further customize the platform.

Reference: Shopify - https://www.shopify.com/

3. Magento: Magento is a robust e-commerce platform that offers advanced customization capabilities. It provides a flexible architecture, allowing you to customize every aspect of your online store. With Magento, you can create custom themes, modify layout templates, and control the appearance of checkout pages, product overviews, and individual product pages. However, Magento might require more technical expertise compared to other platforms due to its complexity.

Reference: Magento - https://magento.com/

4. BigCommerce: BigCommerce is a versatile e-commerce platform that offers a range of customization options. It provides a user-friendly interface and a variety of customizable themes. You can easily modify the layout of checkout pages, product overviews, and individual product pages using its intuitive theme editor. BigCommerce also offers a robust API, allowing you to further customize the platform's functionality.

Reference: BigCommerce - https://www.bigcommerce.com/

Remember, the level of customization also depends on your technical skills and the specific requirements of your project. It's essential to evaluate each platform's features, documentation, and community support to determine which one best fits your needs.