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Here are some particularly nice examples from math function that show real and imaginary parts of special functions on complex axes and on the complex plane.
bessel( n, x ) — the first five Bessel function of the first kind of real or complex order n of a real or complex number.
Bessel functions of the first kind for both positive (blue) and negative (red) real order
Bessel function of the first kind
Bessel function of the second kind
Hankel function of the first kind
Hankel function of the second kind
Jacobi elliptic sine
Jacobi elliptic cosine
Jacobi delta amplitude function
Riemann zeta function
Proper spherical harmonics in 3D
sinc(x,y) in 3D graph
sin(x*y) in 3D graph