  The present paper aims to study the higher-order complete and vertical lifts
of the extended almost complex structures on an extended complex manifold kM.
The proposed theorems on the Nijenhuis tensor of an extended almost complex
structure Jk on the extended complex manifold kM are proved. Also, a tensor
field J~k of type (1,1) is introduced and shows that it is an extended almost
complex structure. Finally, the Lie derivative concerning higher-order lifts is
studied and basic results on the almost analytic complex vector concerning an
extended almost complex structure on kM are investigated.
