Adding simple strafing to my character controller script...

I wonder how this can be achieved in the simplest of matters... as I already said, I want to keep this project simple and organized, limiting everything down and keeping my code clean and organized. This is my modified SimpleMove script from Unity Script Reference:

 using UnityEngine;
 using System.Collections;
 
 [RequireComponent(typeof(CharacterController))]
 
 public class Move : MonoBehaviour {
 
     public float speed = 3.0F;
     public float rotateSpeed = 3.0F;
 
     void Update() {
         CharacterController controller = GetComponent<CharacterController>();
         transform.Rotate(0, Input.GetAxis("Mouse X") * rotateSpeed, 0);
         Vector3 forward = transform.TransformDirection(Vector3.forward);
         float curSpeed = speed * Input.GetAxis("Vertical");
         controller.SimpleMove(forward * curSpeed);
     }
 }

You might've noticed that I am using the mouse to rotate the player... and that is precisely what I want. I do not want an FPS style camera rotation, rather just rotation of the player itself on the horizontal axis.

Thank you all so much in advance! :D