SQL Server 2012 - Create Alias from default instance (i.e. '.') to named instance of SQL 2012
Turns out, it is possible, and after some more experimentation I figured this out. I think the missing elements were: 


Using .\SQL2012 as the server name instead of MACHINENAME\SQL2012
Ensuring SQL Server Browser service is running.


Full Steps


Ensure SQL Server Browser service is started, as well as your named SQL2012 instance.
Open SQL Server Configuration Manager on your machine
Expand the SQL Server Network Configuration node, and double click Protocols for 
Ensure TCP/IP is set to Enabled, if it's not, right-click and choose Enable
Double click TCP/IP


Ensure Listen All is set to Yes
Select the IP Addresses tab, and scroll to the bottom. Make a note of the value in TCP Dynamic Ports.
Press OK

Expand the SQL Native Client 11.0 Configuration node and double click Aliases


Right click in white space and select 'New Alias'
In Alias Name, enter '.'
In Port No, enter the value you noted earlier from TCP Dynamic Ports.
Leave Protocol as TCP/IP
In Server, enter .\&lt;yourinstancename&gt;
Press OK

Repeat the steps for SQL Native Client 11.0 Configuration for SQL Native Client 11.0 Configuration (32 bit)
Test the connection by using SSMS to attempt to connect to '.'