Specifying range in RandomReal

Not with RandomReal that I know of, but you can define a UniformDistribution with your required limits and draw samples from it using RandomVariate:

pts = RandomVariate[UniformDistribution[{{0, 1}, {0, 10}}], 20]

Check that it works as expected:

MinMax /@ Transpose@pts

{{0.00103093, 0.996982}, {0.530043, 9.94596}}

Just marginally faster

x = 1; y = 10; L = 20;
Transpose[{RandomReal[x, L], RandomReal[y, L]}]; // RepeatedTiming

{4.2*10^-6,Null}

ReIm@RandomComplex[{0, x + I y}, L]; // RepeatedTiming

{2.5*10^-6,Null}