Custom fields for Shopify User Registration Form

I can't find any documentation on how to add custom fields for the user registration forum so I tried something but didn't work:

<div class="single-full-width customer">
    <div class="login">
        {% form 'create_customer' %}
            {% if form.errors %}
                {% for field in form.errors %}
                    <p class="input-error-wrapper"><span>{{field}} {{ form.errors.messages[field] }}</span></p>
                {% endfor %}
            {% endif %}
            <p>
                <label>First Name:</label>
                <input type="text" value="" name="customer[first_name]" class="input-text-1" />
            </p>
            <p>
                <label>Last Name:</label>
                <input type="text" value="" name="customer[last_name]" class="input-text-1" />
            </p>
            <p>
                <label>Email Address:</label>
                <input type="text" value="" name="customer[email]" class="input-text-1" />
            </p>

            <p>
                <label>Password:</label>
                <input type="password" value="" name="customer[password]" class="input-text-1" />
            </p>

            <p>
                <label>Pet Name:</label>
                <input type="text" value="" name="customer[pet_name]" class="input-text-1" />

                <input type="radio" name="customer[pet]" value="cat">Cat
                <input type="radio" name="customer[pet]" value="dog">Dog
            </p>

            <p class="sign-in">
                <label></label>
                <a href="#" class="button-1 custom-font-1 trans-1 form-submit-btn"><span>Register</span></a>
                <b>or <a href="{{ shop.url }}">Return to store</a></b>
            </p>
        {% endform %}
    </div>

The fields I've added are the text input - pet name and the radio boxes for Cat/Dog. I've used a test account but the output was that nothing is saved. I've used {{ customer.pet_name }} to try and get the saved value but got nothing in return. Any suggestions? Thanks!

Save your pet name in a tag, or a note since those exist for customers. pet_name obviously is not a customer field.

<p>
  <label>Pet Name:</label>
  <input type="text" value="" name="customer[note]" class="input-text-1" />
  <input type="radio" name="customer[tags]" value="cat">Cat
  <input type="radio" name="customer[tags]" value="dog">Dog
</p>