Are there any non-algebraic, non-transcendental complex numbers? Is $0$ a pure imaginary number

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I came across this question here: Difference between imaginary and complex numbers

The top answer contains this diagram:

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Here we see numbers like $e – \pi i$ and $\pi + 3i$ existing outside of transcendental and algebraic numbers but within the realm of complex numbers. Is this accurate or should they technically be in the transcendental area? Are there any complex non-transcendental non-algebraic numbers?

We also see $0$ as a whole number, an integer, a rational number, a real number, an algebraic number, and a complex number, but is it not also a pure imaginary number?

Best Answer

They're transcendental, but the diagram distinguishes only transcendental real numbers for whatever reason. Non algebraic imaginary numbers are transcendental, and those are represented but not named in the diagram.

$0$ is a real number. Whether or not it's pure imaginary depends on your definition. If you define it as having a real part of $0$, then yes, it is.

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