  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 moduli for SL(2,C)
k-instantons is found to be twice of that of Sp(1), 16k-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 clearify the long standing global
singularity problems associated with Backlund transformations of SU(2)
instantons.
