It's time for a clock challenge!

TI-Basic, 94 bytes

" 
Repeat 99<length(Ans
Ans+"-
End
Ans→Str1
Repeat 0
getTime
ClrDraw
Ans{Ans(1)≠24,1,1
Text(0,0,Ans(1),sub(Str1,1,1+Ans(2
Text(6,0,Ans(1)+1,sub(Str1,1,61-Ans(2
End

Relatively straightforward. That's a string with one space at the beginning. The hours are right-aligned. This only works on TI-84+ calculators since the TI-83 does not have an internal clock.

Edit: Thanks @kundor for noticing that I didn't close the last loop. Fixed now (+2 bytes).

Edit #2: First hour should be zero, not twenty-four. Corrected at a cost of +14 bytes.

Batch, 197 bytes

@echo off
set/ah=100+%time:~0,2%,m=1%time:~3,2%
cls
call:l
set/ah=(h-3)%%24+100,m=260-m
call:l
timeout/t>nul 60
%0
:l
set s=%h:~1% 
for /l %%i in (101,1,%m%)do call set s=%%s%%-
echo %s%

Note: 10th line has a trailing space. For me, %time% formats hours with a leading space but minutes with a leading zero. I decided a leading zero was an easier output format, since all I have to do for that is to add 100 hours and remove the first digit. Minutes are trickier as 08 or 09 will cause octal parse errors, so I prefix a 1 effectively adding 100 minutes, adjusting for this by offsetting the loop appropriately, which is a byte shorter than subtracting the 100.

Python 3.6, 110 114 112 bytes

from time import*
while[sleep(9)]:h,m=localtime()[3:5];print('\n'*50+'%2d '%h+'-'*m+f'\n{-~h%24:2} '+'-'*(60-m))

This uses the new f-string formatting to save one byte (f'\n{h+1:2} ' vs '\n%2d '%(h+1).) You can change [sleep(9)] to 1 to save 8 bytes, but then it just spams the screen.

Saved one byte changing while 1:...;sleep 60 to while[sleep(60)]:..., thanks to TuukkaX.

I had to use 5 more bytes to get the next hour displayed after 23 to be 0, instead of 24, as OP just commented. :-(

Recovered one byte by only sleeping 9 seconds instead of 60.

Saved two bytes using a bit-fiddling to shorten (h+1)%24, borrowed from Value Ink's Ruby answer.