Will inflation rate be the same if we use different base year? I saw this chart where CPI for 2003 is 1.84, for 2004, is 1.889. I dont know the base year because its not in the chart. But base year has to have CPI of 1.

If we instead use 2003 as base year, is the inflation rate during 2003-2004 period, going to be different?
Not at all.
Assuming You wanted to calculate the inflation rate from between 2003 to 2004, you only need to know the CPI for the starting and ending dates.  So the CPI index in 2003 is 1.84 and the CPI index is 1.889 in 2004.

The formula is: (end -start)/start

so we have (1.889-1.84)/1.84 = giving us 0.02663 or a percentage change of 2.663% -This is the inflation rate-.

If we moved the base year to 2003 and assigned the CPI of 1, we can use this calculate the CPI in 2004.

(X-1)/1 = 0.02663
X-1 = 0.02663
X = 1.02663 * 100 is a CPI for 2004 of 102.663.

Made this table for you to get an idea of how it is calculated based on your example.