As discussed here the complex projective space $\mathbb{C}P^n$ is the set of all lines on $\mathbb{C}^n$ passing through the origin. It would seem natural to assume that any $\mathbb{C}P^n$ can be viewed as a tangent space to a point in an equivalently parametrized space. So for a point $p$ on some manifold $M$ the set of all tangent vectors at $p$ is:
$$T_pM$$
As discussed here, the union of all points on a manifold is written as:
$$TM = \bigcup_{p\in M} T_pM$$ and is known as the tangent bundle.
Also, $\pi$ is used to represent the map of from the tangent bundle to the manifold such that $\pi: TM \rightarrow M$ and for each $p \in M$ :
$$\pi^{-1}(p)=T_pM$$
Where $\pi^{-1}(p)$ is a vector space of dimension $n = \dim M$ (confirming as stated above that the tangent space will have same dimension, or equivalent number of parameters as the underlying manifold).
Ordinary spacetime is often described as $\mathbb{R}^{3,1}$ and it is known that euclidean plane $\mathbb{R}^{2}$ can be described by a complex number in $\mathbb{C}^{1}$, and $\mathbb{R}^{1,1}$ can be described with split-complex numbers in $\mathbb{R}^1\oplus \mathbb{R}^1$. When one considers the Lorentz boost, it is tempting to think of ordinary space as being $\mathbb{C}^{1}\oplus \mathbb{R}^1\oplus \mathbb{R}^1$ and described by two variables $\left( z , \mathring{z} \right)$ where $z$ is a complex number and $\mathring{z}$ is a split complex number.
Since there are exactly three $2$-dimensional unital algebras, complex numbers, split-complex numbers and dual numbers, and the algebra of dual numbers is isomorphic to the exterior algebra of $\mathbb{R}^1$, and there are two separate $\mathbb{R}^1$ in $\mathbb{C}^{1}\oplus \mathbb{R}^1\oplus \mathbb{R}^1$, it is tempting to add an additional set of dual numbers so that we can parameterize space as $$\left( z , \mathring{z} \right)\big| \left( \mathring{v} , \mathring{w} \right)$$ where $ \mathring{v}$ is the dual number for the first $\mathbb{R}^1$ and $ \mathring{w} $ is the dual number for the second $\mathbb{R}^1$. Dual numbers are noteworthy as having "fermionic" directions of "bosonic" directions.
If I wanted to keep my new set of coordinates, $$\left( z , \mathring{z} \right)\big| \left( \mathring{v} , \mathring{w} \right)$$ how best would I describe the tangent space of this "ad hoc" manifold? Would I be able to describe this in $\mathbb{C}P^n$, if not, why not? If only because of compactness, why can't tangent spaces be compact? Could we describe it as pseudo-tangent?
