Recently in my algebraic structures class we discussed constructible numbers:
"A real number $\alpha$ if we can construct a line segment of length $|\alpha|$ in a finite number of steps from this given segment of unit length by using a straightedge and a compass." - A First Course in Abstract Algebra, Fraleigh, page 293.
The author goes on to describe all of the wonderful algebraic properties of constructible numbers. Specifically, on page 295 in the proof of Corallary 32.5: "The set of all constructible real numbers forms a subfield F of the field of real numbers," we are given that the unit segment is of length $1 \in \mathbb{R}$.
My question is what if we were to take our unit segment to an element of some other larger field, possibly $\mathbb{C}$ or some field extension of $\mathbb{R}$? For example, if we let our unit segment be $i$, then would it make sense to generate a set of constructible numbers? Would square roots still be present?
I understand this may be somewhat nonsensical, but out of curiosity I was hoping some better versed algebraists could help me explore this line of thought.