millis() uses timer0 (linked to CPU clock) to count time, but ADC_sleep mode stops the CPU clock, therefore millis() will drift (lag behind) after each ADC conversion performed in ADC_sleep mode.

With the standard number of CPU cycles needed for the ADC conversion (ADC prescaler=128 multiplied by ADC clock cycles=13), and with the standard prescaler for timer0 (64), it would look like millis() loses 13.5 tics for every ADC conversion performed in ADC_sleep mode.

I could update the timer0 register (TCNT0) by increasing it manually by 14 but first of all I would still have a 0.5 ticks drift (or a bit less, since the update operation would take some cycles), second I would miss the overflow event (the one Arduino uses internally for time tracking) every time the ADC conversion is started with TCNT0 > 230.

How should I proceed to use ADC_sleep mode during ADC conversions but still keep millis() accurate?