We extend quaternion calculation in the ADHM construction of Sp(1) (=. SU(2)) self-dual Yang-Mills (SDYM) instantons to the case of biquaternion. We use the biconjugate operation of biquaternion first introduced by Hamilton to construct the non-compact SL(2, C). k-instantons. The number of parameters of this construction for SL(2, C). k-instantons is found to be twice of that of Sp(1), 16. k-. 6. These new SL(2, C) instanton solutions contain the SL(2, C). (M, N) instanton solutions constructed previously as a subset. The structures of singularities or jumping lines of the complete SL(2, C). k=. 1, 2, 3 instantons with 10, 26, 42 moduli parameters are particularly investigated. The existence of singular structures of the SL(2, C). k-instantons is mathematically consistent with recent results of solutions of complex ADHM equations. It may also help to clarify the long standing global singularity problems associated with Bäcklund transformations of SU(2) instantons.