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This is really not my expertise: however, some friends and knowledgeable researchers in these fields have shared with me some insight, so I can add my two cents by sharing with you what they shared with me.
The differential and integral calculus for algebras of functions over bicomplex numbers was comprehensively dealt by Griffith Baley Price in his (perhaps classical) monograph [2] on the subject: there he deals also with general multicomplex function algebras and extends the classical Cauchy formulas to those settings.
A more modern exposition, dealing with Cauchy formulas but also with recent progress in the field and also with Stokes and Pompeiu formulas in the context of (only) bicomplex variables is [1] (see in this regard chapter 11, pp. 211-217): it is also a beautiful, smooth, clean and clear, read.
Dealing with differential and integral calculus over this kind of function algebras, both these monographs deal with the concept of differentiability for such functions, and thus give a definition of the (generalized) Cauchy-Riemann equations in these settings.

An historical remark

For general complex function algebras, commutative or not, the classical Cauchy integral theorem was first proved by Giovanni Battista Rizza in 1950: the classical Cauchy integral formula was later extended by him to functions on commutative real normed algebras isomorphic to a complex algebra two years later, in 1952. Many of the details of the researches of Rizza and his school are described in the survey [3].
An addendum. Marin Genov, in his comment below, points out that, earlier than G. B. Rizza, Edgar Raymond Lorch was able to prove a Cauchy formula for infinite dimensional commutative Banach $\Bbb C$-algebras in reference [A1]: this seems the earlier result of this kind, thus I thank Dr. Genov for his reporting.

Bibliography

[1] M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac, Bicomplex holomorphic functions. The algebra, geometry and analysis of bicomplex numbers. (English) Frontiers in Mathematics. Cham: Birkhäuser/Springer, ISBN 978-3-319-24866-0, pp. viii+231 (2015), MR3410909, Zbl 1345.30002.

[2] G. Baley Price, An introduction to multicomplex spaces and functions, (English) Pure and Applied Mathematics, 140. New York etc: Marcel Dekker, Inc. pp. xiii+402 (1991),  ISBN: 0-8247-8345-X, MR1094818, Zbl 0729.30040.

[3] Giovanni Battista Rizza, "Contributi recenti alla teoria delle funzioni nelle algebre", Rendiconti del Seminario Matematico e Fisico di Milano (in Italian), 43 (1): 45–54, doi:10.1007/BF02924838, MR0350025, S2CID 123219540, Zbl 0325.30040.

Addendum reference

[A1] Edgar R. Lorch, "The theory of analytic functions in normed Abelian vector rings", (English), Transaction of the American Mathematical Society 54, pp. 414-425 (1943), DOI:10.2307/1990255, MR0009090, Zbl 0060.27202.

This is really not my expertise: however, some friends and knowledgeable researchers in these fields have shared with me some insight, so I can add my two cents by sharing with you what they shared with me.
The differential and integral calculus for algebras of functions over bicomplex numbers was comprehensively dealt by Griffith Baley Price in his (perhaps classical) monograph [2] on the subject: there he deals also with general multicomplex function algebras and extends the classical Cauchy formulas to those settings.
A more modern exposition, dealing with Cauchy formulas but also with recent progress in the field and also with Stokes and Pompeiu formulas in the context of (only) bicomplex variables is [1] (see in this regard chapter 11, pp. 211-217): it is also a beautiful, smooth, clean and clear, read.
Dealing with differential and integral calculus over this kind of function algebras, both these monographs deal with the concept of differentiability for such functions, and thus give a definition of the (generalized) Cauchy-Riemann equations in these settings.

An historical remark

For general complex function algebras, commutative or not, the classical Cauchy integral theorem was first proved by Giovanni Battista Rizza in 1950: the classical Cauchy integral formula was later extended by him to functions on commutative real normed algebras isomorphic to a complex algebra two years later, in 1952. Many of the details of the researches of Rizza and his school are described in the survey [3].
An addendum. Marin Genov, in his comment below, points out that, earlier than G. B. Rizza, Edgar Raymond Lorch was able to prove a Cauchy formula for infinite dimensional commutative Banach $\Bbb C$-algebras in reference [A1]: this seems the earlier result of this kind, thus I thank Dr. Genov for his reporting.

Bibliography

[1] M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac, Bicomplex holomorphic functions. The algebra, geometry and analysis of bicomplex numbers. (English) Frontiers in Mathematics. Cham: Birkhäuser/Springer, ISBN 978-3-319-24866-0, pp. viii+231 (2015), MR3410909, Zbl 1345.30002.

[2] G. Baley Price, An introduction to multicomplex spaces and functions, (English) Pure and Applied Mathematics, 140. New York etc: Marcel Dekker, Inc. pp. xiii+402 (1991),  ISBN: 0-8247-8345-X, MR1094818, Zbl 0729.30040.

[3] Giovanni Battista Rizza, "Contributi recenti alla teoria delle funzioni nelle algebre", Rendiconti del Seminario Matematico e Fisico di Milano (in Italian), 43 (1): 45–54, doi:10.1007/BF02924838, MR0350025, S2CID 123219540, Zbl 0325.30040.

Addendum reference

[A1] Edgar R. Lorch, "The theory of analytic functions in normed Abelian vector rings", (English), Transaction of the American Mathematical Society 54, pp. 414-425 (1943), DOI:10.2307/1990255, MR0009090, Zbl 0060.27202.

This is really not my expertise: however, some friends and knowledgeable researchers in these fields have shared with me some insight, so I can add my two cents by sharing with you what they shared with me.
The differential and integral calculus for algebras of functions over bicomplex numbers was comprehensively dealt by Griffith Baley Price in his (perhaps classical) monograph [2] on the subject: there he deals also with general multicomplex function algebras and extends the classical Cauchy formulas to those settings.
A more modern exposition, dealing with Cauchy formulas but also with recent progress in the field and also with Stokes and Pompeiu formulas in the context of (only) bicomplex variables is [1] (see in this regard chapter 11, pp. 211-217): it is also a beautiful, smooth, clean and clear, read.
Dealing with differential and integral calculus over this kind of function algebras, both these monographs deal with the concept of differentiability for such functions, and thus give a definition of the (generalized) Cauchy-Riemann equations in these settings.

An historical remark

For general complex function algebras, commutative or not, the classical Cauchy integral theorem was first proved by Giovanni Battista Rizza in 1950: the classical Cauchy integral formula was later extended by him to functions on commutative real normed algebras isomorphic to a complex algebra two years later, in 1952. Many of the details of the researches of Rizza and his school are described in the survey [3].
An addendum. Marin Genov, in his comment below, points out that, earlier than G. B. Rizza, Edgar Raymond Lorch was able to prove a Cauchy formula for infinite dimensional commutative Banach $\Bbb C$-algebras in reference [A1]: this seems the earlier result of this kind, thus I thank Dr. Genov for his reporting.

Bibliography

[1] M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac, Bicomplex holomorphic functions. The algebra, geometry and analysis of bicomplex numbers. (English) Frontiers in Mathematics. Cham: Birkhäuser/Springer, ISBN 978-3-319-24866-0, pp. viii+231 (2015), MR3410909, Zbl 1345.30002.

[2] G. Baley Price, An introduction to multicomplex spaces and functions, (English) Pure and Applied Mathematics, 140. New York etc: Marcel Dekker, Inc. pp. xiii+402 (1991),  ISBN: 0-8247-8345-X, MR1094818, Zbl 0729.30040.

[3] Giovanni Battista Rizza, "Contributi recenti alla teoria delle funzioni nelle algebre", Rendiconti del Seminario Matematico e Fisico di Milano (in Italian), 43 (1): 45–54, doi:10.1007/BF02924838, MR0350025, S2CID 123219540, Zbl 0325.30040.

Addendum reference

[A1] Edgar R. Lorch, "The theory of analytic functions in normed Abelian vector rings", (English), Transaction of the American Mathematical Society 54, pp. 414-425 (1943), DOI:10.2307/1990255, MR0009090, Zbl 0060.27202.

This is really not my expertise: however, some friends and knowledgeable researchers in these fields have shared with me some insight, so I can add my two cents by sharing with you what they shared with me.
The differential and integral calculus for algebras of functions over bicomplex numbers was comprehensively dealt by Griffith Baley Price in his (perhaps classical) monograph [2] on the subject: there he deals also with general multicomplex function algebras and extends the classical Cauchy formulas to those settings.
A more modern exposition, dealing with Cauchy formulas but also with recent progress in the field and also with Stokes and Pompeiu formulas in the context of (only) bicomplex variables is [1] (see in this regard chapter 11, pp. 211-217): it is also a beautiful, smooth, clean and clear, read.
Dealing with differential and integral calculus over this kind of function algebras, both these monographs deal with the concept of differentiability for such functions, and thus give a definition of the (generalized) Cauchy-Riemann equations in these settings.

An historical remark

For general complex function algebras, commutative or not, the classical Cauchy integral theorem was first proved by Giovanni Battista Rizza in 1950: the classical Cauchy integral formula was later extended by him to functions on commutative real normed algebras isomorphic to a complex algebra two years later, in 1952. Many of the details of the researches of Rizza and his school are described in the survey [3].
An addendum. Marin Genov, in his comment below, points out that, earlier than G. B. Rizza, Edgar Raymond Lorch was able to prove a Cauchy formula for infinite dimensional commutative Banach $\Bbb C$-algebras in reference [A1]: this seems the earlier result of this kind, thus I thank Dr. Genov for his reporting.

Bibliography

[1] M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac, Bicomplex holomorphic functions. The algebra, geometry and analysis of bicomplex numbers. (English) Frontiers in Mathematics. Cham: Birkhäuser/Springer, ISBN 978-3-319-24866-0, pp. viii+231 (2015), MR3410909, Zbl 1345.30002.

[2] G. Baley Price, An introduction to multicomplex spaces and functions, (English) Pure and Applied Mathematics, 140. New York etc: Marcel Dekker, Inc. pp. xiii+402 (1991),  ISBN: 0-8247-8345-X, MR1094818, Zbl 0729.30040.

[3] Giovanni Battista Rizza, "Contributi recenti alla teoria delle funzioni nelle algebre", Rendiconti del Seminario Matematico e Fisico di Milano (in Italian), 43 (1): 45–54, doi:10.1007/BF02924838, MR0350025, S2CID 123219540, Zbl 0325.30040.

Addendum reference

[A1] Edgar R. Lorch, "The theory of analytic functions in normed Abelian vector rings", (English), Transaction of the American Mathematical Society 54, pp. 414-425 (1943), DOI:10.2307/1990255, MR0009090, Zbl 0060.27202.

This is really not my expertise: however, some friends and knowledgeable researchers in these fields have shared with me some insight, so I can add my two cents by sharing with you what they shared with me.
The differential and integral calculus for algebras of functions over bicomplex numbers was comprehensively dealt by Griffith Baley Price in his (perhaps classical) monograph [2] on the subject: there he deals also with general multicomplex function algebras and extends the classical Cauchy formulas to those settings.
A more modern exposition, dealing with Cauchy formulas but also with recent progress in the field and also with Stokes and Pompeiu formulas in the context of (only) bicomplex variables is [1] (see in this regard chapter 11, pp. 211-217): it is also a beautiful, smooth, clean and clear, read.
Dealing with differential and integral calculus over this kind of function algebras, both these monographs deal with the concept of differentiability for such functions, and thus give a definition of the (generalized) Cauchy-Riemann equations in these settings.

An historical remark

For general complex function algebras, commutative or not, the classical Cauchy integral theorem was first proved by Giovanni Battista Rizza in 1950: the classical Cauchy integral formula was later extended by him to functions on commutative real normed algebras isomorphic to a complex algebra two years later, in 1952. Many of the details of the researches of Rizza and his school are described in the survey [3].

Bibliography

[1] M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac, Bicomplex holomorphic functions. The algebra, geometry and analysis of bicomplex numbers. (English) Frontiers in Mathematics. Cham: Birkhäuser/Springer, ISBN 978-3-319-24866-0, pp. viii+231 (2015), MR3410909, Zbl 1345.30002.

[2] G. Baley Price, An introduction to multicomplex spaces and functions, (English) Pure and Applied Mathematics, 140. New York etc: Marcel Dekker, Inc. pp. xiii+402 (1991),  ISBN: 0-8247-8345-X, MR1094818, Zbl 0729.30040.

[3] Giovanni Battista Rizza, "Contributi recenti alla teoria delle funzioni nelle algebre", Rendiconti del Seminario Matematico e Fisico di Milano (in Italian), 43 (1): 45–54, doi:10.1007/BF02924838, MR0350025, S2CID 123219540, Zbl 0325.30040.

This is really not my expertise: however, some friends and knowledgeable researchers in these fields have shared with me some insight, so I can add my two cents by sharing with you what they shared with me.
The differential and integral calculus for algebras of functions over bicomplex numbers was comprehensively dealt by Griffith Baley Price in his (perhaps classical) monograph [2] on the subject: there he deals also with general multicomplex function algebras and extends the classical Cauchy formulas to those settings.
A more modern exposition, dealing with Cauchy formulas but also with recent progress in the field and also with Stokes and Pompeiu formulas in the context of (only) bicomplex variables is [1] (see in this regard chapter 11, pp. 211-217): it is also a beautiful, smooth, clean and clear, read.
Dealing with differential and integral calculus over this kind of function algebras, both these monographs deal with the concept of differentiability for such functions, and thus give a definition of the (generalized) Cauchy-Riemann equations in these settings.

An historical remark

For general complex function algebras, commutative or not, the classical Cauchy integral theorem was first proved by Giovanni Battista Rizza in 1950: the classical Cauchy integral formula was later extended by him to functions on commutative real normed algebras isomorphic to a complex algebra two years later, in 1952. Many of the details of the researches of Rizza and his school are described in the survey [3].
An addendum. Marin Genov, in his comment below, points out that, earlier than G. B. Rizza, Edgar Raymond Lorch was able to prove a Cauchy formula for infinite dimensional commutative Banach $\Bbb C$-algebras in reference [A1]: this seems the earlier result of this kind, thus I thank Dr. Genov for his reporting.

Bibliography

[1] M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac, Bicomplex holomorphic functions. The algebra, geometry and analysis of bicomplex numbers. (English) Frontiers in Mathematics. Cham: Birkhäuser/Springer, ISBN 978-3-319-24866-0, pp. viii+231 (2015), MR3410909, Zbl 1345.30002.

[2] G. Baley Price, An introduction to multicomplex spaces and functions, (English) Pure and Applied Mathematics, 140. New York etc: Marcel Dekker, Inc. pp. xiii+402 (1991),  ISBN: 0-8247-8345-X, MR1094818, Zbl 0729.30040.

[3] Giovanni Battista Rizza, "Contributi recenti alla teoria delle funzioni nelle algebre", Rendiconti del Seminario Matematico e Fisico di Milano (in Italian), 43 (1): 45–54, doi:10.1007/BF02924838, MR0350025, S2CID 123219540, Zbl 0325.30040.

Addendum reference

[A1] Edgar R. Lorch, "The theory of analytic functions in normed Abelian vector rings", (English), Transaction of the American Mathematical Society 54, pp. 414-425 (1943), DOI:10.2307/1990255, MR0009090, Zbl 0060.27202.