The study of the \(\mu\)-Darboux transformation of minimal surfaces is based on conformal immersions and Willmore surfaces. First, the authors recall certain notions: generalized Darboux transformations, minimal surfaces, generally conformal immersions of a manifold and Hopf fields. Using an extension of the notion of the associated family \(f_{p,q}\) of a minimal surface to allow quaternionic parameters, the authors show that the pointwise limit of Darboux transformations of \(f:M\to \mathbb{R}^4\) is associated Willmore surface of \(f\) at \(\mu=1\). More precisely, the authors prove the following Theorem: “A conformal immersion \(f:M\to \mathbb{R}^4\), which is not complex holomorphic, is minimal if and only if the kernel of the Hopf field \(A\) is the point at \(\infty\).” Two examples are presented. The subject examined in this paper is interesting. The exposition is clear.