A surface f: \(M^ 2\to X^ 4\) in a 4-dimensional Riemannian manifold \(X^ 4\) is called t-holomorphic if its lift into the twistor space of \(X^ 4\) is holomorphic. Outside the set of all umbilic points a conformally invariant metric on \(M^ 2\) is well-defined. The author studies in case \(X^ 4=R^ 4\) the umbilic points of a t-holomorphic immersion using its general description by a meromorphic function g: \(M^ 2\to {\mathbb{C}}P^ 1\) and two sections \(s_ 1\), \(s_ 2\) in the induced bundle \(g^*(H)\).