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Do the complex numbers form a normal 2D plane?

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What do you mean by "normal" 2D plane?

As a vector space over ℝ, β„‚ is isomorphic with ℝ^(2). Is that what you're asking?
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A complex number a+bi plays a similar role as a R^2 linear transformation which sends the vector (1,0) to (a,b) and the vector (0,1) to (-b,a)

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