Is there a simple way to have a new sidebar for each page?

Registering sidebar is just a function call, there is no reason it can’t be run on top of some logic. Here is code I wrote (and use) in my current theme:

$pages = get_pages();

foreach ( $pages as $page ) {

    register_sidebar( array(
        'name' => "Page: {$page->post_title}",
        'id' => "sidebar-page-{$page->post_name}",
        'description' => "Widgetized area after content of \"{$page->post_title}\" page.",
        'before_widget' => '<div id="%1$s" class="widget %2$s">',
        'after_widget' => '</div>'
        ));
}

Then in page template call sidebar, following same ID convention and you are set.