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Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would like to know if some mathematician has defined a complex order derivative valid without restrictions for all complex number z.

I mean: Is a well-defined definition of such an object possible? I need some refereces about that topic. And related to this question would be the following question: What about a "quaternionic order"/"octonionic order" or maybe a "matricial-order" derivative? Can it be built somehow?Remark: I thought about this extended question when working with the "matrix representation" of complex numbers.

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