Solving n simultaneous differential equation

For small number of ODE's, Mathematica DSolve solves it, but it takes longer time as more ODE's are added.

ClearAll[t, h, b, r, n];
NN = 2;
odes = Table[
  I ToExpression["M" <> ToString[n]]'[t] == 
   b Sqrt[NN + 3 + n]*ToExpression["M" <> ToString[n + 1]][t] + 
    h *Sqrt[n*(2*NN + 5)]*ToExpression["M" <> ToString[n - 1]][t], 
    {n, 0, NN}
];
deps = Table[ToExpression["M" <> ToString[n]][t], {n, 0, NN}] 

Now call DSolve

 DSolve[odes, deps, t]

The solution is too long to post. For N=6 you get

Now it will take much longer time to solve it. I did not want to wait for it.

You did not say how big N is. (btw, N is reserved, so better use different letter)

Edit

To answers comments. To hope to get a solution that can be plotted, need to supply IC and values for the missing parameters $h,b$ and the last $M(t)$. Here is an example for 3 equations just for illustration.

ClearAll[t, h, b, n, M];
NN = 2;
h = 5; b = 6; (*some made up values*)
odes = Table[
  I ToExpression["M" <> ToString[n]]'[t] == 
   b Sqrt[NN + 3 + n]*ToExpression["M" <> ToString[n + 1]][t] + 
    h *Sqrt[n*(2*NN + 5)]*ToExpression["M" <> ToString[n - 1]][t], {n,
    0, NN}] 
deps = Table[ToExpression["M" <> ToString[n]][t], {n, 0, NN}] 
M3[t_] := 2*t; (*some function for the last one, which has no ODE*)
ic = {M0[0] == 1, M1[1] == 2, M2[0] == 2}; (*some IC*)

Now solve the system

DSolve[{odes, ic}, deps, t]

Now the solutions can be plotted. But they are complex. So can plot either the abs or imaginary or real parts. They are complex, since your ODE is complex.

For example

 Plot[Re[M0[t] /. sol], {t, 0, 3}]

 Plot[Re[M1[t] /. sol], {t, 0, 3}]

etc.

Btw, if IC and other values are available, it will be better to use NDSolve instead of DSolve for this. DSolve takes too long time for large N.