z = a + bi   --->   =  =  =     (since  =  =  =    //  <--- since  = -1)

Now you have, from the condition, an equality of these two complex numbers:

 = 5 - 12i.

It means that the real parts are equal and the imaginary parts are equal:

 = 5,     (1)
2ab = -12.       (2)


It is the system of two equations for the two unknowns "a" and "b".
To solve it, express b =  =  from (2) and substitute it into (1). You will get

 = 5,    or

 -  = 5.

Multiply both sides by a^2. You will get

 = 0.

Factor the left side

 = 0.


The equation deploys in two independent equations


1.   = 0  --->   = 9  --->  a = +/- 3  --->  b = -6/a = -/+ 2.


2.   = 0  --->   = -4  --->  No real solutions.


Answer. The solutions of the original equation  = 5 - 12i are these two complex numbers:  = 3 - 2i  or/and   = -3 + 2i.

       In other words, the square root of the complex number 5 - 12i in the complex domain are these two complex numbers: 3 - 2i  and  -3 + 2i.