Can the traditional space of Euclidean vectors (with dot and cross products) be extended to be algebra over reals and what properties would it have?

Question

There is traditional vector calculus of 3D vectors:

• https://en.wikipedia.org/wiki/Euclidean_vector#Basic_properties

• https://en.wikipedia.org/wiki/Vector_algebra_relations

• https://en.wikipedia.org/wiki/Vector_calculus

• https://en.wikipedia.org/wiki/Vector_calculus_identities

In this calculus we usually operate with 3-dimensional vectors and scalars.

But can we consider a sum of a vector and a scalar and define operations on them in such a way so that the vector calculus identities were generalized?

In other words, an algebra of elements of the form

$\mathbf{V}=a_0+a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}=a_0+\mathbf{v}$

There still would be dot and cross products (dot product still always scalar):

$\mathbf{V_1}\cdot \mathbf{V_2}=(a_0+\mathbf{v_1})\cdot(b_0+\mathbf{v_2})=a_0 b_0+\mathbf{v_1}\cdot\mathbf{v_2}$

$\mathbf{V_1}\times \mathbf{V_2}=(a_0+\mathbf{v_1})\times(b_0+\mathbf{v_2})=a_0 \mathbf{v_2}+b_0\mathbf{v_1}+\mathbf{v_1}\times\mathbf{v_2}$

Norm: $||\mathbf{V}||=\sqrt{a_0^2+||\mathbf{v}||^2}=\sqrt{\mathbf{V}\cdot\mathbf{V}}$

And so on.

I wonder, what properties such algebra would have, can we define elementary functions in this space, particularly, logarithms, and how will it compare with hypercomplex numbers.