I would like to implement the function LookRotation() using quaternions, the same one seen in Unity 3D Quaternion.LookRotation(Vector3 forward).

This article almost explains it, at the end is a whole solution but I don't understand the quaternion stuff. I would like to implement it with Quaternion.AngleAxis(). So we would need an angle and rotation vector. But how we can find the rotation vector since LookRotation() only accepts one parameter ?

We could take the cross between forward and Vector3.up to get the right vector. And then cross forward and right to get localUp. But this way we would end up always with a up vector. But we need an arbitrary rotation vector because we need to rotate in all directions not only one (with up vector we can rotate only in one direction depending on left or right hand system).