Is there a reason this does not cause my object to look at mouse pos?

Nothing happens at all.

var position = Input.mousePosition;
	newPosition = Vector3(position.x,position.y,camera.main.transform.position.y- transform.position.y);
	var lastPosition = Camera.main.ScreenToWorldPoint(newPosition);
	transform.LookAt(lastPosition);

Yes. When using Camera.main.ScreenToWorldPoint(), the z parameter in the Vector3 should be set to the distance in front of the camera, not a point in world space. Try using Camera.main.nearClipPlane for z.