No Mathematical Operators Can Phase Me!

$y\color{#3D99F6}{+}y=y \implies 2y=y \implies y=0\\ y\color{#D61F06}-y=y \implies y=0\\ y \color{#20A900}\times y=y \implies y^2-y =0 \implies y(y-1)=0 \implies y=1,\;y=0\\ y \color{#69047E}\div y = y \implies y = 1$

We can see that neither $1$ nor $0$ satisfies this set of equations.

Now, for imaginary numbers, let us try it with the simplest value: $y=i$

$y+y=i+i=2i\neq i$

$y=i$ does not satisfy one of the equations. Therefore, we have sufficient proof to know that not all imaginary numbers satisfy this set of equations.

The answer is $\boxed{\text{None}}$

Relevant wiki: What is 0 divided by 0?

There's no such number that satisfies the conditions. In most of the cases mentioned above in the question "0" does satisfy it for addition, subtraction & multiplication whereas not for division. Because 0/0 is said to be undetermined.

Similarly, in the next case most obvious choice would be 1 for its result in itself for multiplication & division but not for addition & subtraction when and which the answers be 2 & 0 respectively.

Also not all imaginary numbers because for example, if we consider (-1)^(1/2), then it multiplication gives "-1", division does "1", thereby, not satisfying the logic behind. So the answer is "NONE".