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We will see that vectors in $\CC^n$ have natural analogs as rank 2projections in $\RR^{2n}$ and that this association transfers many vectorproperties into properties of rank two projections on $\RR^{2n}$. We believethat this association will answer many open problems in $\CC^n$ where thecorresponding problem in $\RR^n$ has already been answered - and vice versa. Asa application, we will see that phase retrieval (respectively, phase retrievalby projections) in $\CC^n$ transfers to a variation of phase retrieval by rank2 projections (respectively, phase retrieval by projections) on $\RR^{2n}$. Asa consequence, we will answer the open problem: Give the complex version ofEdidin's Theorem \cite{E} which classifies when projections do phase retrievalin $\RR^n$. As another application we answer a longstanding open problemconcerning fusion frames by showing that fusion frames in $\CC^n$ associatewith fusion frames in $\RR^{2n}$ with twice the dimension.  As another application, we will show that a family of mutually unbiased basesin $\CC^n$ has a natural analog as a family of mutually unbiased rank 2projections in $\RR^{2n}$. The importance here is that there are very few realmutually unbiased bases but now there are unlimited numbers of real mutuallyunbiased rank 2 projections to be used in their place. As another application,we will give a variaton of Edidin's theorem which gives a surprisingclassification of norm retrieval. Finally, we will show that equiangular andbiangular frames in $\CC^n$ have an analog as equiangular and biangular rank 2projections in $\RR^{2n}$.

Associating vectors in $\CC^n$ with rank 2 projections in $\RR^{2n}$: with applications