Let \({\mathbb H}=\{ r_0 + r_1i_1 + r_2i_2 + r_3i_3 \mid r_j\in{\mathbb R} \}\) be the skew field of quaternions which satisfy the following relations \[ \begin{aligned} i_1 i_2 &= -i_2 i_1=i_3,\qquad i_2 i_3=-i_3 i_2=i_1,\\ i_3 i_1 &= -i_1 i_3=i_2,\qquad i_1^2=i_2^2=i_3^2=-1.\end{aligned} \] Let \(U\subset{\mathbb H}^n\) be an open domain and \(f:U\to{\mathbb H}\) a smooth function. We may write \(f\) in the form \(f=f_0 + f_1i_1 + f_2i_2 + f_3i_3\), with real smooth functions \(f_j\). The function \(f\) is called \(q\)-holomorphic if it satisfies the differential equation \[ df_0 + i_1 df_1 + i_2 df_2 + i_3 df_3=0. \] This is the analog of the Cauchy-Riemann equations for holomorphic functions. But in contrast to holomorphic functions, the product of two \(q\)-holomorphic functions is in general not \(q\)-holomorphic, in other words, the \(\mathbb H\)-vector space of \(q\)-holomorphic functions is not an algebra. Another problem is that there is no tensor product of \(\mathbb H\)-vector spaces, since \(\mathbb H\) is not commutative. To settle this problem, the author defines a category of augmented \(\mathbb H\)-modules and the tensor product \(V\otimes_{\mathbb H}W\) of two augmented \(\mathbb H\)-modules \(V\) and \(W\). This is again an augmented \(\mathbb H\)-module. Its dimension depends not only on the dimensions of \(V\) and \(W\) but also on their augmentations. An augmented \(\mathbb H\)-module \(A\) together with an \(\mathbb H\)-linear map \(A\otimes_{\mathbb H}A\to A\) is called an \(\mathbb H\)-algebra, if it satisfies certain axioms, like the analog of associativity. In particular, it is shown that the \(\mathbb H\)-vector space of \(q\)-holomorphic functions on \(U\), or more general on a hypercomplex manifold, together with the quaternionic product is an \(\mathbb H\)-algebra. This result is then studied in greater detail and applied to several geometric situations.