Why is my enemy hovering in the air when chasing the player?

Question

In Unity, I have created an enemy AI and applied a script for it to move and chase the player. But when moving, the enemy hovers in the air. My character also does not have a rigid body, he just has character controller and my enemy AI is child object and an empty object as parent object. I added this to do animation. This is my enemy AI script created in C#:

using UnityEngine;
using System.Collections;

public class AdvancedAi : MonoBehaviour

{
    float distance;

    public float lookAtDistance, chaseRange, attackRange, moveSpeed, damping, gravity, attackRepeatTime;
    public Transform target;
    public CharacterController controller;

    private Vector3 moveDirection = Vector3.zero;
    private float attackTime;

    //TODO: add in a function to find controller and to locate and assign the player as the target

    void Start()
    {
        attackTime = Time.time;
    }

    void Update()
    {
        distance = Vector3.Distance(target.position, transform.position);

        if(distance < lookAtDistance)
        {
            LookAt();
        }

        if(distance > lookAtDistance)
        {
            GetComponent<Renderer>().material.color = Color.green;
        }

        if (distance < attackRange)
        {
            AttackPlayer();
        }

        else if (distance < chaseRange)
        {
            ChasePlayer();
        }
    }

    void LookAt()
    {
        GetComponent<Renderer>().material.color = Color.yellow;
        Quaternion rotation = Quaternion.LookRotation(target.position - transform.position);
        transform.rotation = Quaternion.Slerp(transform.rotation, rotation, Time.deltaTime * damping);
    }

    void ChasePlayer()
    {
        GetComponent<Renderer>().material.color = Color.red;

        moveDirection = transform.forward;
        moveDirection *= moveSpeed;

        moveDirection.y -= gravity * Time.deltaTime; // doesn't seem to work right :(
        controller.Move(moveDirection * Time.deltaTime);
    }

    void AttackPlayer()
    {
        //TODO: Need Attack Animations
        if (Time.time > attackTime)
        {
            Debug.Log("Attack");
            attackTime = Time.time + attackRepeatTime;
        }
    }

    void ApplyDamage()
    {
        chaseRange += 30;
        moveSpeed += 2;
        lookAtDistance += 40;
    }
}

Answer 1

CharacterController controller

Has a radius and height that define its space occupation. (the green cage in the image). That cage (capsule) interact with othe colliders (i.e : the terrain). If the cage its too big respect the enemy shape, you see it fluctuate, because the cage bottom touches the ground.

Answer 2

Your todo can be done with, FindGameObjectWihTag ("Player") as there will only ever be one player per instance unless you're doing multi-player.

Damping has no value.

You need a value for gravity on the empty otherwise you have 0 gravity and you're reducing to <0 which would cause it to float.