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So it is well-known that complex differentiability of a function $f:\mathbb{C}\rightarrow\mathbb{C}$ is equivalent to the function being Fréchet/Gateaux differentiable and the component functions (obtained by regarding $\mathbb{C}$ as a 2-dimensional vector space over $\mathbb{R}$) satisfying the Cauchy-Riemann equations, i.e. the Fréchet/Gateaux derivative at $c\in\mathbb{C}$ should be a linear operator representing multiplication by a complex number and thus be of the form $$f'(c) = \begin{pmatrix}a&-b\\b&a\end{pmatrix}.$$

The fact that the "directional derivatives" are required to coincide for all directions in complex analysis leads to a very rigid, yet rich theory. So, related to the above point of view, I wondered if there is a more general theory in which functions on an algebra with Fréchet/Gateaux derivatives that are represented by multiplication operators play an important role? And if so, whether this theory is as rich as complex analysis?

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    $\begingroup$ Interesting question. What if you studied functions $f: \mathbb R^n \to \mathbb R^n$ which are Frechet differentiable and which satisfy $f'(c) = aI + B$ where $a$ is a scalar, $I$ is the identity matrix, and $B$ is skew-symmetric, or something like that. $\endgroup$
    – littleO
    Jun 3, 2021 at 10:25
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    $\begingroup$ The answers to this question I asked some time ago might be of interest to you: math.stackexchange.com/questions/3845538/… Basically, differentiability can also be defined for $p$-adic numbers, though it makes less use of the Fréchet derivative, AFAIK. $\endgroup$ Jun 3, 2021 at 10:28

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