Our Customer Fields app includes an email notification feature so you can be informed each time a new customer registers for an account on your store. The app's form builder also allows you to add a recaptcha, which can be used to prevent fake/spam accounts from being created. Here are some help articles with more details:
A custom app would be simple enough, Shopify API has a callback which can be triggered on a new customer account. You could then trigger an email or a SMS
