An example of where this question presents itself is in the canonical isomorphism between $\mathbb{C}$ and $\mathbb{R}^2$. For $\mathfrak{a},\mathfrak{b}\in\mathbb{C}$ I have been writing
$$\mathfrak{a}^{*}\mathfrak{b}=\mathfrak{a}\cdot\mathfrak{b}+i\mathfrak{a}\wedge\mathfrak{b}.$$
A similar circumstance arises when working with central force problems, such as Kepler orbits. Often all that is needed from a cross product is the signed magnitude of the projection of a vector onto the normal of another. That is, for example
$$v\sin\left[\varphi\right]=\left(\hat{\mathfrak{r}}\times\mathfrak{v}\right)\cdot\hat{\mathfrak{\omega}},$$
gives the scalar component of $\mathfrak{v}$ lying in the orbital plane, and normal to $\hat{\mathfrak{r}}.$ Which is a genuinely cumbersome expression.
I know there are other ways of writing both expressions which do not involve a wedge or cross product. But, the wedge and cross products seem most natural. In the latter, there really is a third dimension which provides an axis for the signed magnitude of the cross product. Nonetheless, once an argument has been made that the problem is essentially two dimensional, that third dimension serves as nothing more than a "bookkeeping" mechanism. It's just a place to put a scalar value.
In the case of the wedge product in $\mathbb{R}^2$ the same sort of thing is happening. We have the bivector $\mathfrak{a}\wedge\mathfrak{b}$ which has only one component. I argue that in both of these cases a mathematically satisfactory argument can be made for writing the desired result as a wedge product, and treating it as a real number.
Another way of saying this is that all $\mathfrak{a}\wedge\mathfrak{b}\backepsilon\mathfrak{a},\mathfrak{b}\in\mathbb{R}^2$ live in a one-dimensional vector space which is isomorphic to $\mathbb{R}$.
So I propose that writing
$$\mathfrak{a}\cdot\mathfrak{b}+i\mathfrak{a}\wedge\mathfrak{b}=ab\left(\cos \varphi+\mathrm{i}\sin\varphi\right),$$
is not merely a convenient mathematical "slang." It is, ineed a completely legitimate mathematical statement.
The argument in support of treating the cross product as a real number is a bit more complicated, since in the case of Kepler orbits, the 3 dimensional interpretation of the cross product has its uses. In the case of $\mathbb{C}\leftrightarrow\mathbb{R}^2$ the cross and wedge products are synonymous. That is, if we even allow for a cross product.
So I ask: is there a valid mathematical argument against treating
$$\mathfrak{a}^{*}\mathfrak{b}=\mathfrak{a}\cdot\mathfrak{b}+i\mathfrak{a}\wedge\mathfrak{b}$$
as a complex number expression?
What about
$$v\sin\left[\varphi\right]=\hat{\mathfrak{r}}\wedge\mathfrak{v}?$$