Trying to make a Floating Enemy follow player

I'm trying to implement an object that will follow behind my player AND rise and fall as my character jumps and land, drops in descent, etc.

The AI is currently using a character controller as collider and is able to follow the player's transform with an offset, but when the player is standing still or jump and matches the AI object's Y, the AI bounces up and down. I am currently using a character controller and characterController.Move(). He is not being affected by gravity.

Here is my current code:

 function Update()
 {
     //If he isn't flinching, he needs to begin his pursuit
     if (activated)
     {
         //Normal AI follows player when in range and attacks when close enough
         if(activated && aiType == "Normal")
             Movement();
         if(activated && aiType == "Observe")
             LookAt();
     }
     
     //Define his movement....
     var movement =  moveDirection  + Vector3 (0, 0, 0);
     
       //...then move him (order of operation :P)
       controller.Move(movement * (Time.deltaTime));
       //transform.position.y = player.transform.position.y;
       
       if (transform.position.y < player.transform.position.y + 2)
       controller.Move(Vector3(0, player.transform.position.y ,0)* (Time.deltaTime));
       else
       if (transform.position.y > player.transform.position.y + 2)
       controller.Move(Vector3(0,-player.transform.position.y ,0)* (Time.deltaTime));
       
       if (transform.position.y == player.transform.position.y + 2)
       controller.Move(Vector3.zero);
 }
 
 
 function LookAt()
 {
    transform.LookAt(Vector3(player.transform.position.x,0,player.transform.position.z));
    transform.eulerAngles = new Vector3(0,transform.eulerAngles.y,0);    
 }
 
 function Movement()
 {
     var direction : Vector3  = player.transform.position - this.transform.position; //target aquired
     
     //If something is in my sights
     if(direction.magnitude < awareness)
     {    
         direction.y = 0; 
     } 
 
     if (direction.magnitude > 1f && activated) //if we are away from the wandering point
     {
         Debug.DrawRay(transform.position+Vector3(0,.1,0), transform.forward, Color.black);
         transform.forward = (transform.forward * 0.9f) + (direction * 0.1f);
         controller.Move((transform.forward * maxSpeed ) * Time.deltaTime);
     }
 } 

Is there a more effective way to make this AI and still retain his solidity?