wp.media.view.ImageDetails – Save settings as HTML5 data-* attributes for image

A way to do it is to use the (very convenient) editor:image-edit and editor:image-update events triggered by the tinymce wpeditimage plugin to get/set the dom directly (updated to wrap in wp_enqueue_media action):

add_action( 'wp_enqueue_media', function () {
    add_action( 'admin_footer', function () {
        ?>
        <script type="text/javascript">
        jQuery(function ($) {
            if (wp && wp.media && wp.media.events) {
                wp.media.events.on( 'editor:image-edit', function (data) {
                    data.metadata.my_setting = data.editor.dom.getAttrib( data.image, 'data-my_setting' );
                } );
                wp.media.events.on( 'editor:image-update', function (data) {
                    data.editor.dom.setAttrib( data.image, 'data-my_setting', data.metadata.my_setting );
                } );
            }
        });
        </script>
        <?php
    }, 11 );
} );

To add and populate the settings field, it’s perhaps groovier to hack the output of wp_print_media_templates() rather than override ImageDetails.initialize() (updated to wrap in wp_enqueue_media action):

add_action( 'wp_enqueue_media', function () {
    remove_action( 'admin_footer', 'wp_print_media_templates' );
    add_action( 'admin_footer', $func = function () {
        ob_start();
        wp_print_media_templates();
        $tpl = ob_get_clean();
        // To future-proof a bit, search first for the template and then for the section.
        if ( ( $idx = strpos( $tpl, 'tmpl-image-details' ) ) !== false
                && ( $before_idx = strpos( $tpl, '<div class="advanced-section">', $idx ) ) !== false ) {
            ob_start();
            ?>
    <div class="my_setting-section">
        <h2><?php _e( 'My Settings' ); ?></h2>
        <div class="my_setting">
            <label class="setting my_setting">
                <span><?php _e( 'My Setting' ); ?></span>
                <input type="text" data-setting="my_setting" value="{{ data.model.my_setting }}" />
            </label>
        </div>
    </div>
            <?php
            $my_section = ob_get_clean();
            $tpl = substr_replace( $tpl, $my_section, $before_idx, 0 );
        }
        echo $tpl;
    } );
} );