Is it possible to set templates at the user level?
If not then is there a plugin?
If not what can I do?
The reason is that I want to build a new template for an existing website and see the build on my user account.
Rather than working on your live site, you should develop your new template on a copy of your site. Using Akeeba Backup and Kickstart, it's usually a question of minutes to create an exact copy of your website. It's both safer and easier than working on your live site.
If you still want to test your new template on your live site, you can change it adding
?template=youTemplateName to your url:
You could also try an extension called "Template Selector" from JED. It's a module that let's you change between the available templates, and if you set the module Access parameter to "Super users", it will only be visible to (super)administrators.