Solve the equation (z+i)^*=2zi+1.

STEP 1: What the questions asks us is to find z that solves the equation given. Since z is a complex number, we need to determine both its real and imaginary parts. Hence, we begin by writing z in terms of its real and imaginary parts, z = a+bi (notice the imaginary part is whatever is multiplied by the number "i").
STEP 2: Using substitution, this gives (z+i) = a+bi+i. It is also useful to group the real and imaginary parts, in order to get (z+i) = a+(b+1)i.
STEP 3: Express the complex conjugate of (z+i) as (z+i)* = a-(b+1)i.
STEP 4: Substitute everything in original equation, to obtain the following equality: a-(b+1)i = (1-2b)+2ai.
STEP 5: Equate real and imaginary parts to get a system of equations in a and b: a = 1-2b and -(b+1) = 2a
STEP 6: Solve, to obtain a = -1, and b = 1. This gives z = -1+i.