Note the "many" in the reference page for NDSolve, under More Information:
"NDSolve can solve many differential-algebraic equations, in which some of the eqns are purely algebraic, or some of the variables are implicitly algebraic."
The equation starting with Sqrt[z[t]] is algebraic. The warning message says the method NDSolve uses for differential-algebraic equations handles real values only.
The equation starting with Sqrt[z[t]] is also wrong at t=0, since you get 0 == 339.583 I.
Can you re-arrange the equations to have just differential equations? For example, Solve the algebraic equation for Z[t], take the derivative, and include that as a third differential equation in the system.
For backing up a calculation that is going the wrong way (like taking T[t] negative) see: