Designing a non linear observer often requires to immerse the system dynamics into a space of greater dimension. In order to avoid the difficult left inversion of the immersion, coordinate augmentation has been proposed in [9]. However, difficulties in this augmentation via continuous Jacobian completion, losses of observability, or the question of completeness of solutions may arise in practice. This paper illustrates with the help of a toy-system how to overcome them by using tools and tricks such as extending the image of a diffeomorphism or adding fictitious outputs.