MISSING_ARGUMENT, Id not specified in an update call in controller extension

Question

I'm having some difficulty in setting the value for a custom object from my controller extension. Here's the relevant controller code:

public with sharing class MyController {
    public ApexPages.StandardController stdController { get; set; }

    public AxonifyRedirectController(ApexPages.StandardController controller) {
        stdController = controller;
    }

    public PageReference save() {
        Custom__c customObject = (Custom__c) stdController.getRecord();
        update customObject;

        return null;
    }
}

The custom object is a hierarchy type set to protected visibility.

The VisualForce page looks this:

<apex:page sidebar="false" standardController="Custom__c" extensions="MyController">
    <apex:outputPanel id="myPanel"">
        <apex:form>
            <apex:pageBlock title="Custom Object Setup">
                Please enter your custom value.<p/>
                <apex:inputField value="{!Custom__c.Code__c}" /><p/>
                <apex:commandButton action="{!save}" value="Save" />
            </apex:pageBlock>
        </apex:form>
    </apex:outputPanel>
</apex:page>

I'm sure it's something ridiculously simple, but I just can't find the right documentation.