Enemy follow script problem

Hey guys, this might be a really simple question but I just can’t seem to figure it out, I have a follow script attached to my “Enemies” and it works, they rotate towards the player and follow him till they are a certain distance away.

However, the enemies seem to move along the y axis to match the players height (resulting in the enemies partialy going throught the terrain) so I changed it to follow the main cam which is a first person cam attached to the player (now the enemies are floating above ground to match the cams height) so my question is how do I stop the gameobjects from moving along the y axis? No rigidbodies are attached to enemies just a trigger and a collider. Here’s my script; p.s have tried with rigidbodies and constrants, gravity, etc.

public Transform player;
	public float moveSpeed = 5f;
	public float minDist = 2f;


	// Use this for initialization
	void Start () {
	
		player = GameObject.FindGameObjectWithTag ("MainCamera").transform;
	}
	
	// Update is called once per frame
	void Update () {
	
		transform.LookAt (player);
		if (Vector3.Distance (transform.position, player.position) >= minDist) {
		
			transform.position += transform.forward * moveSpeed * Time.fixedDeltaTime;
		}

	}
}

//if enemy must go in y axis
if (Vector3.Distance (transform.position, player.position) >= minDist) {

         transform.position += transform.up* moveSpeed * Time.fixedDeltaTime;
     }

also check enemy rotation to see what axis is he r…
if blue line (Z) is looking at Top so your script is ok

The simplest and straightforward solution that comes to mind is to just use the x and z components, like this:

// at beginning of class:
private Vector3 displacement;

// Update or whatever you use:
displacement = transform.forward * moveSpeed * Time.fixedDeltaTime; 
transform.position += new Vector3(displacement.x, 0, displacement.z) ;

or be done with it by using:

transform.position += Vector3.Scale(transform.forward * moveSpeed * Time.fixedDeltaTime, new Vector3(1, 0, 1)) ;