Movement along X and Z axis...

I'm slightly confused, do you want your character to spin as if it were turning to face your cursor? If so, that means that you want it to rotate only on the "Y" axis. I might be able to help, but I need more information.

Wow I've not explained it properly :P

Yeah is pretty much what you've said, I want one of its sides to always face the cursor, hence using only the Y axis. If you can help me with that, it'll be great :D

If you just replace the (lookPos.y, lookPos.x) * Mathf.Rad2Deg;

With (lookPos.z, lookPos.x) * Mathf.Rad2Deg;

Then it should rotate along the x and z and not the y.

So something like this:

using UnityEngine; using System.Collections;

 public class MouseLook : MonoBehaviour {
  
 void Update () {
            Vector3 mousePos = new Vector3(Input.mousePosition.x, Input.mousePosition.z, 10);
            Vector3 lookPos = Camera.main.ScreenToWorldPoint(mousePos);
            lookPos = lookPos - transform.position;
            float angle = Mathf.Atan2(lookPos.z, lookPos.x) * Mathf.Rad2Deg;
            transform.rotation = Quaternion.AngleAxis(angle, Vector3.forward);
  
         }
     }

Let me know if that works:D

@Destran's answer is close, you also have to replace the Vector3.forward with 'Vector3.up' in the last line.

   transform.rotation = Quaternion.AngleAxis(angle, Vector3.up);

It's closer than the others so far... However I still need it to face the cursor whilst it moves in 360 degrees. With this script + robertbu's contribution, it only rotates half-way, and mirrors that for the other half.