The existence of such an immersion is related to the existence of odd Hopf invariant in $\pi_7(S^4)$. Namely take any such element, for example the homotopy class of the Hopf map. Represent it - by the Pontrjagin construction - as an embedded framed 3-dimensional submanifold in $R^7$. Using the so called Compression theorem (by Rourke Sanderson, it is more or less equivalent to Smale Hirsch immersion theory) you may isotope this framed immersion so, that one of the framing normal vectors becomes everywhere parallel to the 7-th coordinate direction of $R^7$. Then projecting it to $R^6$ we obtain a framed immersion in $R^6$. The (algebraic) number of double points (they have signs, since the double branches have an ordering, which was higher, which was lower) agrees with the Hopf invariant of the original map. Applying two more times the compression, you can push the immersion into $R^4$ as an immersion. When you lift back from $R^4$ to $R^6$ after arbitrary regular homotopy, the pairity of the double points remains unchanged. (This is the stable Hopf invariant of the element of the stable homotopy class in $\pi^s(3)$ represented by the obtained immersion.)