Should the cursor be visible?

It is entirely possible to implement a custom cursor instead of the system one. To do this you would hide the system one, track mouse position or movement and display your own image in the needed place.

	// Hide the cursor
	Screen.showCursor = false;

using UnityEngine;
using System.Collections;

public class Example : MonoBehaviour {
    void Example() {
        Screen.showCursor = false;
    }
}

import UnityEngine
import System.Collections

public class Example(MonoBehaviour):

	def Example() as void:
		Screen.showCursor = false