Very nice color demos!
Since we can use EXIT FOR, why not speed up Julia.bas with a FOR loop and EXIT FOR? I've also removed repeated calculations to increase the speed further.
So I suggest the following change to the "Loop processing" part:
MODE 3
CLS
'Specify initial values
RealOffset = -1.30
ImaginOffset = -1.22
'------------------------------------------------*
'Set the Julia set constant [eg C = -1.2 + 0.8i]
CRealVal = -0.78
CImagVal = -0.20
'------------------------------------------------*
MAXIT=80 'max iterations
PixelWidth = MM.HRES
PixelHeight = MM.VRES
GAP = PixelHeight / PixelWidth
SIZE = 2.50
XDelta = SIZE / PixelWidth
YDelta = (SIZE * GAP) / PixelHeight
'Loop processing - visit every pixel
FOR X = 0 TO (PixelWidth - 1)
CX = X * Xdelta + RealOffset
FOR Y = 0 TO (PixelHeight - 1)
CY = Y * YDelta + ImaginOffset
Zr = CX
Zi = CY
COUNT = 0
'Begin Iteration loop
FOR COUNT = 0 TO MAXIT-1
Zr2 = Zr*Zr: Zi2 = Zi*Zi
IF Zr2+Zi2 >= 4 THEN EXIT FOR
Zi = 2*Zr*Zi+CImagVal: Zr = Zr2-Zi2+CRealVal
NEXT COUNT
PIXEL(X,Y) = COUNT MOD 8
NEXT Y
NEXT X
DO
a$ = INKEY$
LOOP WHILE a$ = ""