Hypercomplex numbers (a+bi with a,b hyperreal)

[lolgarithms]

Does the field of "hypercomplex numbers" (*C; a+bi with a,b hyperreal numbers) satisfy the transfer principle? Are all of the arithmetical facts with complex numbers true with the hypercomplex numbers?

One might be able to construct a nonstandard complex analysis with the hypercomplex numbers, with identical results as as standard complex analysis, with the standard part function st(z) = st(Re(z)) + i st(Im(z)) used to define complex derivatives and integrals.

 

[g_edgar]

Yes, there is no problem doing what you want. I learned nonstandard analysis from Robinson's book, which is still a classic. From that approach, you see how it is not limited to real numbers, but encompasses all of analysis.

P.S. The term "hypercomplex numbers" is used more commonly with a completely different meaning. For what you want, maybe say "complex numbers in nonstandard analysis".