I believe that what you want is to create a 3D model from a series of photographs of your subjects head. The process of doing this is technically a form of photogrammetry
This can be done reasonably cheaply and simply now days via taking a series of photos at different angles of your model (with a single hand held camera), and then feeding the individual photos into software that then creates the model from those images.
The final model then can be rotated and zoomed as desired in various 3d model editors.
Here is a random blog that discusses the process. I found this blog by googling "create 3d models from photos" (note I am not affiliated with this blog - and there are a zillion other websites that show similar techniques)
The one caveat is that because you are building a 3D model, you'll have to know something about the 3D software itself and be able to do things like clean up artifacts that can arise because the model is stitched together from non-contiguous photos.
Another item you may want to google is "360 photography turntables" (and related subjects). These are used to rotate an object while you take photos from a fixed camera location. I once saw a documentary of a photographer going to comic-cons and using such a setup to take 360 degree photos of people in costume.
And finally, if you have $$$ to spare you can investigate "multi-camera rigs" that surround your subject with cameras and take all the required photos at a single instant. This will help with ensuring you have correctly aligned and lit images, but you'll still have to go through the same stitching process as with a single camera.