  Based on the quaternionic approach developed by one of us [Z.D. Zhang, Phil.
Mag. 87 (2007) 5309.] for the three-dimensional (3D) Ising model, we study in
this work conformal invariance in three dimensions. We develop a procedure for
treating the 3D conformal field theory. The 2D conformal field theory is
generalized to be appropriate for three dimensions, within the framework of
quaternionic coordinates with complex weights. The Virasoro algebra still
works, but for each complex plane of quaternionic coordinates. The quaternionic
geometric phases appear in quaternionic Hilbert space as a result of
diagonalization procedure which involves the smoothing of knots/crossings in
the 3D many-body interacting spin Ising system. Possibility for application of
conformal invariance in three dimensions on studying the behaviour of the world
volume of the brane, or the world sheet of the string in 3D or (3+1)D, is
briefly discussed.
