I prefer a simpler perspective for complex numbers of "defined latently, then discovered, accepted, named and given notation", other than "invented".
Invented implies some degree of arbitrariness or choice, but complex numbers are not an arbitrary construct.
Zero, negative numbers, and imaginary numbers were all latently defined by prior concepts before they were recognized. They were unavoidable, as existing operations inevitably kept producing them. Since they kept coming up, it forced people to eventually recognize that these seemingly nonsensical concepts continued to behave sensibly under the operations that produced them.
Once addition and subtraction were defined on natural numbers, (1, 2, 3, ... etc), the concept of zero was latently defined. The concept of "nothing" was not immediately recognized as a number, but there is only one consistent way of dealing with 2-2, 5-5, 7-7, etc. Eventually that concept was given a name "zero", notation "0", and adopted as a number.
It was discovered, in that it was already determined by addition and subtraction, just not yet recognized.
Similarly with negative numbers. They were also latently determined by addition and subtraction. At first subtracting a larger number from a smaller number was considered nonsensical. But starting from the simple acceptance that "5-8" can at least be consistently viewed as the number which added to 8 gives 5, and other similar examples, it was discovered that such numbers had only one consistent behavior.
So they were accepted, given a name "negative numbers" and a notation "-x", short hand for "0-x".
And again, once addition, multiplication, (and optionally exponentiation) were defined, the expressions x*x = -1 (or x = sqrt(-1)) were run into, they were initially considered non-sensical.
But starting from acceptance that it at least makes sense to say that "the square of the square root of -1", is "-1", it was discovered that roots of -1 could be worked with consistently using the already accepted operations that produced them.
The numbers that included square roots of -1 were given a name "imaginary numbers", the square root of -1 given notation, "i", and we got complex numbers that had both real and square root of -1 parts.
Invented implies some degree of arbitrariness or choice, but complex numbers are not an arbitrary construct.
Zero, negative numbers, and imaginary numbers were all latently defined by prior concepts before they were recognized. They were unavoidable, as existing operations inevitably kept producing them. Since they kept coming up, it forced people to eventually recognize that these seemingly nonsensical concepts continued to behave sensibly under the operations that produced them.
Once addition and subtraction were defined on natural numbers, (1, 2, 3, ... etc), the concept of zero was latently defined. The concept of "nothing" was not immediately recognized as a number, but there is only one consistent way of dealing with 2-2, 5-5, 7-7, etc. Eventually that concept was given a name "zero", notation "0", and adopted as a number.
It was discovered, in that it was already determined by addition and subtraction, just not yet recognized.
Similarly with negative numbers. They were also latently determined by addition and subtraction. At first subtracting a larger number from a smaller number was considered nonsensical. But starting from the simple acceptance that "5-8" can at least be consistently viewed as the number which added to 8 gives 5, and other similar examples, it was discovered that such numbers had only one consistent behavior.
So they were accepted, given a name "negative numbers" and a notation "-x", short hand for "0-x".
And again, once addition, multiplication, (and optionally exponentiation) were defined, the expressions x*x = -1 (or x = sqrt(-1)) were run into, they were initially considered non-sensical.
But starting from acceptance that it at least makes sense to say that "the square of the square root of -1", is "-1", it was discovered that roots of -1 could be worked with consistently using the already accepted operations that produced them.
The numbers that included square roots of -1 were given a name "imaginary numbers", the square root of -1 given notation, "i", and we got complex numbers that had both real and square root of -1 parts.