x^2 + 1 = 0?

x² = - 1
x² = i ²
x = ± i (two imaginary numbers)

x^2 + 1 = 0
x^2 =-1
x=+/-sqr(-1) which can not be

so the solution is " 2 imaginary numbers "

The basic building block of imaginary numbers is: i
i = (-1) ^ (1/2)

Given this:
x^2 + 1 = 0
x^2 = -1
x =+/-(-1)^(1/2)
Since the basic building block of imaginary numbers is DEFINED as (-1)^(1/2) and negative i is just as imaginary the answer is obviously...

x^2 + 1 = 0
x^2 = -1
x = root (-1)
since the root of a negative number cannot be taken, the answer is 2 imaginary numbers.

The other answers tell you the actual results, but you can also consider the discriminant (b^2 -4ac)

a=1, b=0, c=1

so b^2 - 4ac = 0^2 - 4x1x1 = -4

This is negative, so x^2 +1 = 0 must have no real roots, so in your list of options, that means 2 imaginary roots.

two imaginary numbers as x is the square root of -1 which is not a real number

i and -i

x^2 + 1 = 0
x^2 = -1
x = ±√-1 (2 imaginary numbers)

Two imaginary numbers
sqrt(-1) = i , -i

where i is an imaginay root named iota